Angle Relationships5-1
Vocabularyangle adjacent angles
right angle supplementary angles
acute angle complementary angles
obtuse angle
straight angle
vertical angles
congruent angles
Angle Relationships5-1
An angle () is formed by two rays, or sides, with a common endpoint called the vertex.
*You can name an angle several ways: 1) by its vertex
2)by its vertex and a point on each ray 3) by a number.
*When three points are used, the middle point must be the vertex.
Angle Relationships5-1
Additional Example 1: Classifying Angles
A. two acute angles
B. two obtuse angles
SQP, RQT
TQP, RQS
Use the diagram to name each figure.
mTQP = 43°; mRQS = 47°
mSQP= 133°; mRQT = 137°
Angle Relationships5-1
Additional Example 1: Classifying Angles
C. a pair of complementary angles
B. two pairs of supplementary angles
TQP, TQR
TQP, RQS
Use the diagram to name each figure.
mTQP + mRQS = 43° + 47° = 90
mTQP + mTQR = 43° + 137° = 180
SQP, SQR mSQP + mSQR = 133° + 47° = 180
Angle Relationships5-1Check It Out: Example 1
A. two acute angles
B. two obtuse angles
AEC, BED
AEB, CED
Use the diagram to name each figure.
mAEB = 15°; mCED = 75°
mAEC= 105°; mBED = 165°
Angle Relationships5-1Check It Out: Example 1
C. a pair of complementary angles
D. a pair of supplementary angles
CED, AEC
AEB, CED mAEB + mCED= 15° + 75° = 90
mCED + mAEC = 75° + 105° = 180
Use the diagram to name each figure.
Angle Relationships5-1
Additional Example 2A: Finding Angle Measures
Use the diagram to find each angle measure.
If m1 = 37°, find m2.
1 and 2 are supplementary.
Substitute 37 for m1.
m1 + m2 = 180°
37° + m2= 180°
m2 = 143°
–37° –37° Subtract 37 from both sides.
Angle Relationships5-1
Additional Example 2B: Finding Angle Measures
Use the diagram to find each angle measure.
Find m3, if m<2= 143°.
2 and 3 are supplementary.
Substitute 143 for m2.
m2 + m3 = 180°
143° + m3 = 180°
m3 = 37°
–143° –143° Subtract 143 from both sides.
Angle Relationships5-1
Check It Out: Example 2
Use the diagram to find each angle measure.
If m1 = 42°, find m2.
1 and 2 are supplementary.
Substitute 42 for m1.
m1 + m2 = 180°
42° + m2= 180°
m2 = 138°
–42° –42° Subtract 42 from both sides.
Angle Relationships5-1
Adjacent angles have a common vertex and a common side, but no common interior points. Angles 1 and 2 in the diagram are adjacent angles.
Congruent angles have the same measure.
Vertical angles are the nonadjacent angles formed by two intersecting lines. Angles 2 and 4 are vertical angles. Vertical angles are congruent.
Angle Relationships5-1
Additional Example 3: Application
A traffic engineer designed a section of roadway where three streets intersect. Based on the diagram, what is the measure of DBE.Step 1: Find mCBD.
Vertical angles are congruent.ABF CBD
mABF = mCBD
mCBD = 26
Congruent angles have the same measure.Substitute 26 for mCBD.
Angle Relationships5-1
Additional Example 3 Continued
A traffic engineer designed a section of roadway where three streets intersect. Based on the diagram, what is the measure of DBE.Step 2: Find mDBE.
The angles are complementary.
Substitute 26 for mCBD.
mCBD + mDEB = 90°
26 + mDEB = 90°
mDEB = 64°
–26° –26° Subtract 26 from both sides.
Angle Relationships5-1
Check It Out: Example 3
A traffic engineer designed a section of roadway where three streets intersect. Based on the diagram, what is the measure of DBE.Step 1: Find mCBD.
Vertical angles are congruent.ABF CBD
mABF = mCBD
mCBD = 19
Congruent angles have the same measure.Substitute 19 for mCBD.
19
Angle Relationships5-1
Check It Out: Example 3 Continued
A traffic engineer designed a section of roadway where three streets intersect. Based on the diagram, what is the measure of DBE.Step 2: Find mDBE.
The angles are complementary.
Substitute 19 for mCBD.
mCBD + mDEB = 90°
19 + mDEB = 90°
mDEB = 71°
–19° –19° Subtract 19 from both sides.
19