Right Angle Theorem Lesson 4.3

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.

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.

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

Given: ABCD is a rhombus AB  BC  CD  AD Prove: AC BD 5 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