 TEKS Focus:  (6)(A) Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angle formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems.

 What is an Angle? What is an Angle?

An angle is a figure formed by two rays, or sides, with the same endpoint called the vertex (plural: vertices). You can name an angle several ways: by its vertex, by a point on each ray and the vertex, or by a number. B A C 1 Names:  B  ABC  CBA  1 side

 The measure of an angle is the absolute value of the difference of the real numbers paired with the sides of the angle.

 What are Acute, Obtuse, Right, and Straight Angles? What are Acute, Obtuse, Right, and Straight Angles?

Congruent angles are angles that have the same measure. In the diagram, mABC = mDEF, so you can write ABC  DEF. This is read as “angle ABC is congruent to angle DEF.” Arc marks are used to show that the two angles are congruent.

The set of all points between the sides of the angle is the interior of an angle. The exterior of an angle is the set of all points outside the angle. There is more than one way to name an angle: Angle Name could be R, SRT, TRS, or 1

Write the different ways you can name the angles in the diagram. QTS, QTR, STR, 1, 2 Example: 1

Use the diagram to find the measure of each angle. Then classify each as acute, right, or obtuse. a. BOA mBOA = 40° mDOB = 125°mDOC = 90° BOA is acute. DOB is obtuse.EOC is right. Example: 2 b. DOBc. DOC

mDEG = 115°, and mDEF = 48°. Find mFEG mDEG = mDEF + mFEG 115 = 48 + mFEG 67 = mFEG  Add. Post. Substitute the given values. Subtract 48 from both sides. Simplify. –48° Example: 3

mXWZ = 121°. m XWY = (4x + 6)°. mYWZ = (6x – 5)°. Find mXWY and mYWZ. mXWZ = mXWY + mYWZ 121 = (4x + 6) + (6x – 5) 121 = 10x = 10x x = 12 mXWY = 54 and mYWZ = 67  Add. Post. Substitute the given values. Combine like terms Subtraction Prop. of = Division Prop. of = Example: 4

JKM  MKL, mJKM = (4x + 6)°, and mMKL = (7x – 12)°. Find mJKM. Example: 5

Step 1 Find x. mJKM = mMKL (4x + 6)° = (7x – 12)° +12 4x + 18 = 7x –4x 18 = 3x 6 = x Definition of congruent angles. Substitute the given values. Add 12 to both sides. Simplify. Subtract 4x from both sides. Divide both sides by 3. Simplify. Example: 5 continued Step 2 Find mJKM. mJKM = 4x + 6 = 4(6) + 6 = 30 Substitute 6 for x. Simplify.

  RQT is a straight angle.  What are m  TQS and  m  RQS? 6x x + 4 = 180 8x + 24 = 180 8x = 156 x = 19.5 m  TQS = 6(19.5) + 20 = 137  and m  RQS = 2(19.5) + 4 = 43  Def. of straight  ;  Add. Post. Combine like terms. Subtraction Property of Equality Division Property of Equality Substitution Property of Equality

 EXTRA EXAMPLES NOT USED IN COMPOSITION BOOK FOLLOW

A surveyor recorded the angles formed by a transit (point A) and three distant points, B, C, and D. Name three of the angles. Possible answer: BAC CAD BAD Example: 7 Can you name it  A? Why or why not? No; adjacent angles cannot be named by just the vertex.

Find the measure of each angle. Then classify each as acute, right, or obtuse. A. WXVB. ZXW mWXV = 30° WXV is acute. mZXW = |130° - 30°| = 100° ZXW = is obtuse. Example: 8

Find the measure of each angle. mLJK = mLJM. Given mLJK = (-10x + 3)°, and mKJM = (–x + 21)°. Find mLJM. mLJK = mKJM (–10x + 3)° = (–x + 21)° –9x + 3 = 21 x = –2 Step 1 Find x. –9x = 18 +x +x –3 Substitute the given values. Add x to both sides. Simplify. Subtract 3 from both sides. Divide both sides by –9. Simplify. Example: 9

Step 2 Find mLJM. mLJM = mLJK + mKJM = (–10x + 3)° + (–x + 21)° = –10(–2) + 3 – (–2) + 21Substitute –2 for x. Simplify.= = 46° Example: 9 continued