![Angles,[object Object],[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-1-2048.jpg?cb=1667922606)
![§1.6 Perpendicular Lines,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-7-2048.jpg?cb=1667922606)
![Z,[object Object],Y,[object Object],XY and XZ are ____________.,[object Object],X,[object Object],Angles,[object Object],Opposite rays,[object Object],___________ are two rays that are part of a the same line and have only theirendpoints in common.,[object Object],opposite rays,[object Object],straight angle,[object Object],The figure formed by opposite rays is also referred to as a ____________.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-8-2048.jpg?cb=1667922606)
![S,[object Object],vertex,[object Object],T,[object Object],Angles,[object Object],There is another case where two rays can have a common endpoint.,[object Object],angle,[object Object],This figure is called an _____.,[object Object],Some parts of angles have special names.,[object Object],The common endpoint is called the ______,,[object Object],vertex,[object Object],and the two rays that make up the sides ofthe angle are called the sides of the angle.,[object Object],side,[object Object],R,[object Object],side,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-9-2048.jpg?cb=1667922606)
![S,[object Object],vertex,[object Object],SRT,[object Object],TRS,[object Object],R,[object Object],1,[object Object],T,[object Object],Angles,[object Object],There are several ways to name this angle.,[object Object],1) Use the vertex and a point from each side. ,[object Object],or,[object Object],The vertex letter is always in the middle.,[object Object],side,[object Object],2) Use the vertex only.,[object Object],1,[object Object],If there is only one angle at a vertex, then theangle can be named with that vertex.,[object Object],R,[object Object],side,[object Object],3) Use a number.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-10-2048.jpg?cb=1667922606)
![D,[object Object],2,[object Object],F,[object Object],DEF,[object Object],2,[object Object],E,[object Object],FED,[object Object],E,[object Object],Angles,[object Object],Symbols:,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-11-2048.jpg?cb=1667922606)
![C,[object Object],A,[object Object],1,[object Object],B,[object Object],ABC,[object Object],1,[object Object],B,[object Object],CBA,[object Object],BA and,[object Object],BC,[object Object],Angles,[object Object],1) Name the angle in four ways.,[object Object],2) Identify the vertex and sides of this angle.,[object Object],vertex:,[object Object],Point B,[object Object],sides:,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-12-2048.jpg?cb=1667922606)
![2) What are other names for ?,[object Object],3) Is there an angle that can be named ? ,[object Object],1,[object Object],2,[object Object],1,[object Object],XWY or,[object Object],YWX,[object Object],W,[object Object],XWZ,[object Object],Angles,[object Object],1) Name all angles having W as their vertex.,[object Object],X,[object Object],W,[object Object],1,[object Object],2,[object Object],Y,[object Object],Z,[object Object],No!,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-13-2048.jpg?cb=1667922606)
![exterior,[object Object],W,[object Object],Y,[object Object],Z,[object Object],interior,[object Object],A,[object Object],B,[object Object],Angles,[object Object],An angle separates a plane into three parts:,[object Object],interior,[object Object],1) the ______,[object Object],exterior,[object Object],2) the ______,[object Object],angle itself ,[object Object],3) the _________,[object Object],In the figure shown, point B and all other points in the blue region are in the interiorof the angle.,[object Object],Point A and all other points in the greenregion are in the exterior of the angle.,[object Object],Points Y, W, and Z are on the angle.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-14-2048.jpg?cb=1667922606)
![P,[object Object],G,[object Object],Angles,[object Object],Is point B in the interior of the angle, exterior of the angle, or on the angle?,[object Object],Exterior,[object Object],B,[object Object],Is point G in the interior of the angle, exterior of the angle, or on the angle?,[object Object],On the angle,[object Object],Is point P in the interior of the angle, exterior of the angle, or on the angle?,[object Object],Interior,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-15-2048.jpg?cb=1667922606)
![Vocabulary,[object Object],§1.6 Angle Measure,[object Object],What You'll Learn,[object Object],You will learn to measure, draw, and classify angles.,[object Object],1) Degrees,[object Object],2) Protractor,[object Object],3)Right Angle,[object Object],4) Acute Angle,[object Object],5) Obtuse Angle,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-16-2048.jpg?cb=1667922606)
![P,[object Object],75°,[object Object],Q,[object Object],R,[object Object],m PQR = 75,[object Object],§1.6 Angle Measure,[object Object],degrees,[object Object],In geometry, angles are measured in units called _______.,[object Object],The symbol for degree is °.,[object Object],In the figure to the right, the angle is 75 degrees.,[object Object],In notation, there is no degree symbol with 75,[object Object],because the measure of an angle is a real ,[object Object],number with no unit of measure.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-17-2048.jpg?cb=1667922606)
![A,[object Object],n°,[object Object],C,[object Object],m ABC = n,[object Object],and 0 < n < 180,[object Object],B,[object Object],§1.6 Angle Measure,[object Object],0,[object Object],180,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-18-2048.jpg?cb=1667922606)
![Use a protractor to measure SRQ.,[object Object],1) Place the center point of the protractor on vertex R. Align the straightedge with side RS.,[object Object],2) Use the scale that begins with 0 at RS. Read where the other side of the angle, RQ, crosses this scale.,[object Object],Q,[object Object],R,[object Object],S,[object Object],§1.6 Angle Measure,[object Object],protractor,[object Object],You can use a _________ to measure angles and sketch angles of givenmeasure.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-19-2048.jpg?cb=1667922606)
![m SRQ = ,[object Object],m SRJ = ,[object Object],m SRG = ,[object Object],m QRG = ,[object Object],m GRJ = ,[object Object],m SRH ,[object Object],H,[object Object],J,[object Object],G,[object Object],Q,[object Object],S,[object Object],R,[object Object],§1.6 Angle Measure,[object Object],70,[object Object],Find the measurement of:,[object Object], 180 – 150,[object Object],= 30,[object Object],180,[object Object],45,[object Object], 150 – 45,[object Object],= 105,[object Object],150,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-20-2048.jpg?cb=1667922606)
![1) Draw AB,[object Object],3) Locate and draw point C at the mark labeled 135. Draw AC.,[object Object],C,[object Object],A,[object Object],B,[object Object],§1.6 Angle Measure,[object Object],Use a protractor to draw an angle having a measure of 135.,[object Object],2) Place the center point of the protractor on A. Align the mark labeled 0 with the ray.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-21-2048.jpg?cb=1667922606)
![A,[object Object],A,[object Object],A,[object Object], obtuse angle 90 < m A < 180,[object Object],acute angle 0 < m A < 90,[object Object],right angle m A = 90,[object Object],§1.6 Angle Measure,[object Object],Once the measure of an angle is known, the angle can be classified as oneof three types of angles. These types are defined in relation to a right angle.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-22-2048.jpg?cb=1667922606)
![40°,[object Object],110°,[object Object],90°,[object Object],50°,[object Object],75°,[object Object],130°,[object Object],§1.6 Angle Measure,[object Object],Classify each angle as acute, obtuse, or right.,[object Object],Acute,[object Object],Obtuse,[object Object],Right,[object Object],Obtuse,[object Object],Acute,[object Object],Acute,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-23-2048.jpg?cb=1667922606)
![The measure of H is 67.Solve for y.,[object Object],The measure of B is 138.Solve for x.,[object Object],H ,[object Object],9y + 4,[object Object],5x - 7,[object Object],B ,[object Object],B = 5x – 7 and B = 138,[object Object],H = 9y + 4 and H = 67,[object Object],§1.6 Angle Measure,[object Object],Given: (What do you know?),[object Object],Given: (What do you know?),[object Object],9y + 4 = 67,[object Object],5x – 7 = 138,[object Object],Check!,[object Object],Check!,[object Object],9y = 63,[object Object],5x = 145,[object Object],9(7) + 4 = ?,[object Object],5(29) -7 = ?,[object Object],y = 7,[object Object],x = 29,[object Object],63 + 4 = ?,[object Object],145 -7 = ?,[object Object],67 = 67,[object Object],138 = 138,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-24-2048.jpg?cb=1667922606)
![Is m a larger than m b ?,[object Object],? ? ?,[object Object],60°,[object Object],60°,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-25-2048.jpg?cb=1667922606)
![End of Lesson,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-26-2048.jpg?cb=1667922606)
![Vocabulary,[object Object],§1.6 The Angle Addition Postulate,[object Object],What You'll Learn,[object Object],You will learn to find the measure of an angle and the bisectorof an angle. ,[object Object],NOTHING NEW!,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-27-2048.jpg?cb=1667922606)
![R,[object Object],X,[object Object],2) Draw and label a point X in the interior of the angle. Then draw SX.,[object Object],S,[object Object],T,[object Object],§1.6 The Angle Addition Postulate,[object Object],1) Draw an acute, an obtuse, or a right angle. Label the angle RST.,[object Object],45°,[object Object],75°,[object Object],30°,[object Object],3) For each angle, find mRSX, mXST, and RST.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-28-2048.jpg?cb=1667922606)
![R,[object Object],X,[object Object],S,[object Object],T,[object Object],§1.6 The Angle Addition Postulate,[object Object],1) How does the sum of mRSX and mXST compare to mRST ?,[object Object],Their sum is equal to the measure of RST .,[object Object],mXST = 30,[object Object],+ mRSX = 45,[object Object],= mRST = 75,[object Object],2) Make a conjecture about the relationship between the two smaller angles and the larger angle.,[object Object],45°,[object Object],The sum of the measures of the twosmaller angles is equal to the measureof the larger angle.,[object Object],75°,[object Object],30°,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-29-2048.jpg?cb=1667922606)
![P,[object Object],1,[object Object],Q,[object Object],A,[object Object],2,[object Object],R,[object Object],§1.6 The Angle Addition Postulate,[object Object],m1 + m2 = mPQR.,[object Object],There are two equations that can be derived using Postulate 3 – 3.,[object Object],m1 = mPQR –m2 ,[object Object],These equations are true no matter where A is locatedin the interior of PQR. ,[object Object],m2 = mPQR –m1 ,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-30-2048.jpg?cb=1667922606)
![X,[object Object],1,[object Object],Y,[object Object],W,[object Object],2,[object Object],Z,[object Object],§1.6 The Angle Addition Postulate,[object Object],Find m2 if mXYZ = 86 and m1 = 22.,[object Object],Postulate 3 – 3.,[object Object],m2 + m1 = mXYZ,[object Object],m2 = mXYZ –m1 ,[object Object],m2 = 86 – 22,[object Object],m2 = 64,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-31-2048.jpg?cb=1667922606)
![C,[object Object],D,[object Object],(5x – 6)°,[object Object],2x°,[object Object],B,[object Object],A,[object Object],§1.6 The Angle Addition Postulate,[object Object],Find mABC and mCBD if mABD = 120.,[object Object],mABC + mCBD = mABD,[object Object],Angle Addition Postulate,[object Object],Substitution,[object Object],2x + (5x – 6) = 120,[object Object],7x – 6 = 120,[object Object],Combine like terms,[object Object],7x = 126,[object Object],Add 6 to both sides,[object Object],x = 18,[object Object],Divide each side by 7,[object Object],36 + 84 = 120,[object Object],mCBD = 5x – 6 ,[object Object],mABC = 2x,[object Object],mCBD = 5(18) – 6 ,[object Object],mABC = 2(18),[object Object],mCBD = 90 – 6 ,[object Object],mABC = 36,[object Object],mCBD = 84 ,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-32-2048.jpg?cb=1667922606)
![§1.6 The Angle Addition Postulate,[object Object],Just as every segment has a midpoint that bisects the segment, every angle,[object Object],has a ___ that bisects the angle.,[object Object],ray,[object Object],angle bisector,[object Object],This ray is called an ____________ .,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-33-2048.jpg?cb=1667922606)
![is the bisector of PQR.,[object Object],P,[object Object],1,[object Object],Q,[object Object],A,[object Object],2,[object Object],R,[object Object],§1.6 The Angle Addition Postulate,[object Object],m1 = m2,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-34-2048.jpg?cb=1667922606)
![Since bisects CAN, 1 = 2.,[object Object],N,[object Object],T,[object Object],2,[object Object],1,[object Object],A,[object Object],C,[object Object],§1.6 The Angle Addition Postulate,[object Object],If bisects CAN and mCAN = 130, find 1 and 2.,[object Object],1 + 2 = CAN,[object Object],Angle Addition Postulate,[object Object],Replace CAN with 130,[object Object],1 + 2 = 130,[object Object],1 + 1 = 130,[object Object],Replace 2 with 1,[object Object],2(1) = 130,[object Object],Combine like terms,[object Object],(1) = 65,[object Object],Divide each side by 2,[object Object],Since 1 = 2, 2 = 65,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-35-2048.jpg?cb=1667922606)
![End of Lesson,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-36-2048.jpg?cb=1667922606)
![A,[object Object],B,[object Object],D,[object Object],C,[object Object],Adjacent Angles and Linear Pairs of Angles,[object Object],What You'll Learn,[object Object],You will learn to identify and use adjacent angles and linear pairs of angles.,[object Object],When you “split” an angle, you create two angles. ,[object Object],The two angles are called,[object Object], _____________,[object Object],adjacent angles,[object Object],2,[object Object],1,[object Object],adjacent = next to, joining.,[object Object],1 and 2 are examples of adjacent angles. They share a common ray.,[object Object],Name the ray that 1 and 2 have in common. ____,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-37-2048.jpg?cb=1667922606)
![J,[object Object],2,[object Object],common side,[object Object],R,[object Object],M,[object Object],1,[object Object],1 and 2 are adjacent,[object Object],with the same vertex R and,[object Object],N,[object Object],Adjacent Angles and Linear Pairs of Angles,[object Object],Adjacent angles are angles that:,[object Object],A) share a common side,[object Object],B) have the same vertex, and,[object Object],C) have no interior points in common,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-38-2048.jpg?cb=1667922606)
![B,[object Object],2,[object Object],1,[object Object],1,[object Object],2,[object Object],G,[object Object],N,[object Object],L,[object Object],1,[object Object],J,[object Object],2,[object Object],Adjacent Angles and Linear Pairs of Angles,[object Object],Determine whether 1 and 2 are adjacent angles.,[object Object],No. They have a common vertex B, but,[object Object], _____________,[object Object],no common side,[object Object],Yes. They have the same vertex G and a common side with no interior points in common.,[object Object],No. They do not have a common vertex or ____________,[object Object],a common side,[object Object],The side of 1 is ____,[object Object],The side of 2 is ____,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-39-2048.jpg?cb=1667922606)
![1,[object Object],2,[object Object],1,[object Object],2,[object Object],Z,[object Object],D,[object Object],X,[object Object],Adjacent Angles and Linear Pairs of Angles,[object Object],Determine whether 1 and 2 are adjacent angles.,[object Object],No. ,[object Object],Yes. ,[object Object],In this example, the noncommon sides of the adjacent angles form a,[object Object],___________.,[object Object],straight line,[object Object],linear pair,[object Object],These angles are called a _________,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-40-2048.jpg?cb=1667922606)
![D,[object Object],A,[object Object],B,[object Object],2,[object Object],1,[object Object],C,[object Object],Note:,[object Object],Adjacent Angles and Linear Pairs of Angles,[object Object],Two angles form a linear pair if and only if (iff):,[object Object],A) they are adjacent and,[object Object],B) their noncommon sides are opposite rays,[object Object],1 and 2 are a linear pair.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-41-2048.jpg?cb=1667922606)
![In the figure, and are opposite rays.,[object Object],H,[object Object],T,[object Object],E,[object Object],3,[object Object],A,[object Object],4,[object Object],2,[object Object],1,[object Object],C,[object Object],ACE and 1 have a common side ,,[object Object],the same vertex C, and opposite rays,[object Object], and,[object Object],M,[object Object],Adjacent Angles and Linear Pairs of Angles,[object Object],1) Name the angle that forms a ,[object Object], linear pair with 1.,[object Object],ACE,[object Object],2) Do 3 and TCM form a linear pair? Justify your answer.,[object Object],No. Their noncommon sides are not opposite rays.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-42-2048.jpg?cb=1667922606)
![End of Lesson,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-43-2048.jpg?cb=1667922606)
![§1.6 Complementary and Supplementary Angles,[object Object],What You'll Learn,[object Object],You will learn to identify and use Complementary and ,[object Object],Supplementary angles,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-44-2048.jpg?cb=1667922606)
![E,[object Object],D,[object Object],A,[object Object],60°,[object Object],30°,[object Object],F,[object Object],B,[object Object],C,[object Object],§1.6 Complementary and Supplementary Angles,[object Object],Two angles are complementary if and only if (iff) the sum of their degree measure is 90. ,[object Object],mABC + mDEF = 30 + 60 = 90,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-45-2048.jpg?cb=1667922606)
![E,[object Object],D,[object Object],A,[object Object],60°,[object Object],30°,[object Object],F,[object Object],B,[object Object],C,[object Object],§1.6 Complementary and Supplementary Angles,[object Object],If two angles are complementary, each angle is a complement of the other.,[object Object],ABC is the complement of DEF and DEF is the complement of ABC.,[object Object],Complementary angles DO NOT need to have a common side or even the ,[object Object],same vertex.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-46-2048.jpg?cb=1667922606)
![I,[object Object],75°,[object Object],15°,[object Object],H,[object Object],P,[object Object],Q,[object Object],40°,[object Object],50°,[object Object],H,[object Object],S,[object Object],U,[object Object],V,[object Object],60°,[object Object],T,[object Object],30°,[object Object],Z,[object Object],W,[object Object],§1.6 Complementary and Supplementary Angles,[object Object],Some examples of complementary angles are shown below.,[object Object],mH + mI = 90,[object Object],mPHQ + mQHS = 90,[object Object],mTZU + mVZW = 90,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-47-2048.jpg?cb=1667922606)
![D,[object Object],C,[object Object],130°,[object Object],50°,[object Object],E,[object Object],B,[object Object],F,[object Object],A,[object Object],§1.6 Complementary and Supplementary Angles,[object Object],If the sum of the measure of two angles is 180, they form a special pair of ,[object Object],angles called supplementary angles.,[object Object],Two angles are supplementary if and only if (iff) the sum of their degree measure is 180. ,[object Object],mABC + mDEF = 50 + 130 = 180,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-48-2048.jpg?cb=1667922606)
![I,[object Object],75°,[object Object],105°,[object Object],H,[object Object],Q,[object Object],130°,[object Object],50°,[object Object],H,[object Object],S,[object Object],P,[object Object],U,[object Object],V,[object Object],60°,[object Object],120°,[object Object],60°,[object Object],Z,[object Object],W,[object Object],T,[object Object],§1.6 Complementary and Supplementary Angles,[object Object],Some examples of supplementary angles are shown below.,[object Object],mH + mI = 180,[object Object],mPHQ + mQHS = 180,[object Object],mTZU + mUZV = 180,[object Object],and,[object Object],mTZU + mVZW = 180,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-49-2048.jpg?cb=1667922606)
![End of Lesson,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-50-2048.jpg?cb=1667922606)
![§1.6 Congruent Angles,[object Object],What You'll Learn,[object Object],You will learn to identify and use congruent and,[object Object],vertical angles.,[object Object],Recall that congruent segments have the same ________.,[object Object],measure,[object Object],Congruent angles,[object Object],_______________ also have the same measure.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-51-2048.jpg?cb=1667922606)
![50°,[object Object],50°,[object Object],B,[object Object],V,[object Object],§1.6 Congruent Angles,[object Object],Two angles are congruent iff, they have the same,[object Object],______________.,[object Object],degree measure,[object Object],B V iff,[object Object],mB = mV,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-52-2048.jpg?cb=1667922606)
![1,[object Object],2,[object Object],X,[object Object],Z,[object Object],§1.6 Congruent Angles,[object Object],arcs,[object Object],To show that 1 is congruent to 2, we use ____.,[object Object],To show that there is a second set of congruent angles, X and Z, we use double arcs.,[object Object],This “arc” notation states that:,[object Object],X Z,[object Object],mX = mZ,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-53-2048.jpg?cb=1667922606)
![§1.6 Congruent Angles,[object Object],four,[object Object],When two lines intersect, ____ angles are formed.,[object Object],There are two pair of nonadjacent angles.,[object Object],vertical angles,[object Object],These pairs are called _____________.,[object Object],1,[object Object],4,[object Object],2,[object Object],3,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-54-2048.jpg?cb=1667922606)
![§1.6 Congruent Angles,[object Object],Two angles are vertical iff they are two nonadjacent,[object Object],angles formed by a pair of intersecting lines.,[object Object],Vertical angles:,[object Object],1 and 3,[object Object],1,[object Object],4,[object Object],2,[object Object],2 and 4,[object Object],3,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-55-2048.jpg?cb=1667922606)
![1,[object Object],4,[object Object],2,[object Object],3,[object Object],§1.6 Congruent Angles,[object Object],1) On a sheet of paper, construct two intersecting lines,[object Object], that are not perpendicular.,[object Object],2) With a protractor, measure each angle formed.,[object Object],3) Make a conjecture about vertical angles.,[object Object],Consider:,[object Object],A. 1 is supplementary to 4.,[object Object],m1 + m4 = 180,[object Object],Hands-On,[object Object],B. 3 is supplementary to 4.,[object Object],m3 + m4 = 180,[object Object],Therefore, it can be shown that,[object Object],1 3,[object Object],Likewise, it can be shown that,[object Object],24,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-56-2048.jpg?cb=1667922606)
![1,[object Object],4,[object Object],2,[object Object],3,[object Object],§1.6 Congruent Angles,[object Object],1) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3,[object Object],x = 4; 3 = 19°,[object Object],2) If m2 = x + 9 and the m3 = 2x + 3, then find the m4,[object Object],x = 56; 4 = 65°,[object Object],3) If m2 = 6x - 1 and the m4 = 4x + 17, then find the m3,[object Object],x = 9; 3 = 127°,[object Object],4) If m1 = 9x - 7 and the m3 = 6x + 23, then find the m4,[object Object],x = 10; 4 = 97°,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-57-2048.jpg?cb=1667922606)
![§1.6 Congruent Angles,[object Object],Vertical angles are congruent.,[object Object],n,[object Object],m,[object Object],2,[object Object],1 3,[object Object],3,[object Object],1,[object Object],2 4,[object Object],4,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-58-2048.jpg?cb=1667922606)
![130°,[object Object],x°,[object Object],§1.6 Congruent Angles,[object Object],Find the value of x in the figure:,[object Object],The angles are vertical angles.,[object Object],So, the value of x is 130°.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-59-2048.jpg?cb=1667922606)
![§1.6 Congruent Angles,[object Object],Find the value of x in the figure:,[object Object],The angles are vertical angles.,[object Object],(x – 10) = 125.,[object Object],(x – 10)°,[object Object],x – 10 = 125.,[object Object],125°,[object Object],x = 135.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-60-2048.jpg?cb=1667922606)
![§1.6 Congruent Angles,[object Object],Suppose two angles are congruent.,[object Object],What do you think is true about their complements?,[object Object],1 2,[object Object],2 + y = 90,[object Object],1 + x = 90,[object Object],y is the complement ,[object Object],of 2,[object Object],x is the complement ,[object Object],of 1,[object Object],y = 90 - 2,[object Object],x = 90 - 1,[object Object],Because 1 2, a “substitution” is made.,[object Object],y = 90 - 1,[object Object],x = 90 - 1,[object Object],x = y,[object Object],x y,[object Object],If two angles are congruent, their complements are congruent.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-61-2048.jpg?cb=1667922606)
![60°,[object Object],60°,[object Object],B,[object Object],A,[object Object],1,[object Object],2,[object Object],3,[object Object],4,[object Object],§1.6 Congruent Angles,[object Object],If two angles are congruent, then their complements are,[object Object],_________.,[object Object],congruent,[object Object],The measure of angles complementary to A and B,[object Object],is 30.,[object Object],A B,[object Object],If two angles are congruent, then their supplements are,[object Object],_________.,[object Object],congruent,[object Object],The measure of angles supplementary to 1 and 4,[object Object],is 110.,[object Object],110°,[object Object],110°,[object Object],70°,[object Object],70°,[object Object],4 1,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-62-2048.jpg?cb=1667922606)
![3,[object Object],1,[object Object],2,[object Object],§1.6 Congruent Angles,[object Object],If two angles are complementary to the same angle,,[object Object],then they are _________.,[object Object],congruent,[object Object],3 is complementary to 4,[object Object],5 is complementary to 4,[object Object],4,[object Object],3,[object Object],5,[object Object],5 3,[object Object],If two angles are supplementary to the same angle,,[object Object],then they are _________.,[object Object],congruent,[object Object],1 is supplementary to 2,[object Object],3 is supplementary to 2,[object Object],1 3,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-63-2048.jpg?cb=1667922606)
![52°,[object Object],52°,[object Object],A,[object Object],B,[object Object],§1.6 Congruent Angles,[object Object],Suppose A B and mA = 52.,[object Object],Find the measure of an angle that is supplementary to B.,[object Object],1,[object Object],B + 1 = 180,[object Object],1 = 180 – B,[object Object],1 = 180 – 52,[object Object],1 = 128°,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-64-2048.jpg?cb=1667922606)
![§1.6 Congruent Angles,[object Object],If 1 is complementary to 3,,[object Object], 2 is complementary to 3,,[object Object], and m3 = 25,,[object Object], What are m1 and m2 ?,[object Object],m1 + m3 = 90 Definition of complementary angles.,[object Object],m1 = 90 - m3 Subtract m3 from both sides.,[object Object],m1 = 90 - 25Substitute 25 in for m3.,[object Object],m1 = 65Simplify the right side.,[object Object],You solve for m2,[object Object],m2 + m3 = 90 Definition of complementary angles.,[object Object],m2 = 90 - m3 Subtract m3 from both sides.,[object Object],m2 = 90 - 25Substitute 25 in for m3.,[object Object],m2 = 65Simplify the right side.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-65-2048.jpg?cb=1667922606)
![G,[object Object],D,[object Object],1,[object Object],2,[object Object],A,[object Object],C,[object Object],4,[object Object],B,[object Object],3,[object Object],E,[object Object],H,[object Object],§1.6 Congruent Angles,[object Object],1) If m1 = 2x + 3 and the m3 = 3x - 14, then find the m3,[object Object],x = 17; 3 = 37°,[object Object],2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC,[object Object],x = 29; EBC = 121°,[object Object],3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4,[object Object],x = 16; 4 = 39°,[object Object],4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1,[object Object],x = 18; 1 = 43°,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-66-2048.jpg?cb=1667922606)
![Suppose you draw two angles that are congruent and supplementary.,[object Object],What is true about the angles?,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-67-2048.jpg?cb=1667922606)
![1,[object Object],2,[object Object],C,[object Object],A,[object Object],B,[object Object],§1.6 Congruent Angles,[object Object],If two angles are congruent and supplementary then each is a __________.,[object Object],right angle,[object Object],1 is supplementary to 2,[object Object],1 and 2 = 90,[object Object],All right angles are _________.,[object Object],congruent,[object Object],A B C,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-68-2048.jpg?cb=1667922606)
![B,[object Object],A,[object Object],2,[object Object],E,[object Object],3,[object Object],1,[object Object],4,[object Object],C,[object Object],D,[object Object],§1.6 Congruent Angles,[object Object],If 1 is supplementary to 4, 3 is supplementary to 4, and,[object Object],m 1 = 64, what are m 3 and m 4?,[object Object],They are vertical angles.,[object Object],1 3,[object Object],m 1 = m3,[object Object],m 3 = 64,[object Object],3 is supplementary to 4,[object Object],Given,[object Object],Definition of supplementary.,[object Object],m3 + m4 = 180,[object Object],64 + m4 = 180,[object Object],m4 = 180 – 64,[object Object],m4 = 116,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-69-2048.jpg?cb=1667922606)
![End of Lesson,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-70-2048.jpg?cb=1667922606)
![§1.6 Perpendicular Lines,[object Object],What You'll Learn,[object Object],You will learn to identify, use properties of, and construct,[object Object],perpendicular lines and segments.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-71-2048.jpg?cb=1667922606)
![In the figure below, lines are perpendicular.,[object Object],A,[object Object],1,[object Object],2,[object Object],C,[object Object],D,[object Object],4,[object Object],3,[object Object],B,[object Object],§1.6 Perpendicular Lines,[object Object],perpendicular lines,[object Object],Lines that intersect at an angle of 90 degrees are _________________.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-72-2048.jpg?cb=1667922606)
![m,[object Object],n,[object Object],§1.6 Perpendicular Lines,[object Object],Perpendicular lines are lines that intersect to form a,[object Object],right angle.,[object Object]](https://image.slidesharecdn.com/1-7anglesandperpendicularlines-100916082437-phpapp02/75/17-angles-and-perpendicular-lines-73-2048.jpg?cb=1667922606)
1.7 angles and perpendicular lines
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1.7 angles and perpendicular lines
- 3. §1.6The Angle Addition Postulate
- 4. §1.6Adjacent Angles and Linear Pairs of Angles
- 5. §1.6Complementary and Supplementary Angles