1. Right Angle Theorem
Lesson 4.3

2. Theorem 23: If two angles are both supplementary and congruent, then they are right angles.1 2
Given: 1  2
Prove: 1 and 2 are right angles.

3. Paragraph Proof:
Since 1 and 2 form a straight angle, they are supplementary. Therefore, m1 + m2 = 180°.
Since 1 and 2 are congruent, we can use substitution to get the equation:
m1 + m2 = 180° or m1 = 90°.
Thus, 1 is a right angle and so is 2.

4. Given: Circle P S is the midpoint of QR Prove: PS QR P Τ Q S R
Draw PQ and PR
PQ  PR
S mdpt QR
QS  RS
PS  PS
PSQ  PSR
PSQ  PSR
QSR is a straight 
PSQ & PSR are supp.
PSQ and PSR are rt s
PS QR
Given
Two points determine a seg.
Radii of a circle are  .
A mdpt divides a segment into 2  segs.
Reflexive property.
SSS
CPCTC
Assumed from diagram.
2 s that make a straight  are supp.
If 2 s are both supp and , they are rt s.
If 2 lines intersect to form rt s, they are . Τ Τ

5. Given: ABCD is a rhombus AB  BC  CD  AD Prove: AC BD5 4 7 2 E 1 Τ 3 6 8 B C
Hint: Draw and label shape!
Given
Reflexive Property
SSS
CPCTC
If then
ASA
Assumed from diagram.
2 s that make a straight  are supp.
If 2 s are both supp and  they are rt s.
If 2 lines intersect and form rt s, they are .
AB  BC  CD  AD
AC  AC
BAC  DAC
7  5
3  4
ABE  ADE
1  2
BED is a straight 
1 & 2 are supp.
1 and 2 are rt s
AC BD Τ Τ