1. Warm up Draw two congruent angles: Draw two adjacent angles:
Draw two angles that are complementary:
Draw an angle which also has a ray that bisects the angle:

2. Proving Statements about Segments and Angles
Chapter 2.5

3. We learned these in Chapter 1!
To recap (again..):
2 column proof: has numbered STATEMENTS on one side and corresponding REASONS on the other
So far, we have proved statements using algebra operations and properties.
Today we prove statements using geometric properties
We learned these in Chapter 1!

4. Adding to our proof reason list (purple sheet)
Segment Addition Postulate
Segment Congruence Property (reflexive, symmetric, transitive)
Angle Congruence Property (reflexive, symmetric, transitive)
m <PQR = m <PQS + m<SQR
Angle Addition Postulate

5. Congruent / Equal “Congruent” is different from “equal” M L K J
Things that are EQUAL have a numeric value, like the measure of an angle in degrees, or the length of a line segment.
m <ABC = 30⁰
LM = 45 cm L M
45 cm
30⁰
Shapes are CONGRUENT, things like angles, line segments, and polygons. Two shapes are congruent when they have the same measure.
m<ABC = 30⁰ and m<XYZ = 30⁰, so <ABC and <XYZ are congruent
LM = 45 cm and JK = 45 cm
So LM and JK are congruent
30⁰ x y z J K
45 cm

6. Properties of Congruence

7. Properties of Congruence

8. Example.. 1. m < 2 = 145⁰ 2. < 1 and < 2 are supplementary
3. m < 1 + m < 2 = 180⁰
4. m < ⁰ = 180⁰
5. m < 1 = 35⁰
1. Given
2. Given
3. Definition of Supplementary Angles
4. Substitution Property of Equality
5. Subtraction Property of Equality

9. Example 1. < 1 is a complement of < 2
2. <2 is congruent to < 3
3. m < 1 + m < 2 = 90⁰
4. m < 2 = m < 3
5. m < 1 + m < 3 = 90⁰
6. < 1 is a complement of < 3
1. Given
2. Given
3. Definition of Complementary Angles
4. Definition of congruent angles
5. Substitution Property of Equality
6. Definition of Complementary Angles

10. Example Solve for x. Justify each step. 1. QR ≅ PQ 2. RS ≅ PQ
3. QR ≅ RS
4. QR = RS
5. (2x + 5) = (10 – 3x)
6. 5x + 5 = 10
7. 5x = 5
8. x = 1
1. Given
2. Given
3. Transitive Prop Equality
4. Definition of congruent segments
5. Substitution
6. Addition Prop Equality
7. Subtraction Prop Equality
8. Division Prop Equality