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, m1 + m2 = 180°.

Since 1 and 2 are congruent, we can use substitution to get the equation:

m1 + m2 = 180° or m1 = 90°.

Thus, 1 is a right angle and so is 2.

P

Q RS

Given: Circle P

S is the midpoint of QR

Prove: PS QR

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1. Circle P2. Draw PQ and PR3. PQ PR4. S mdpt QR5. QS RS6. PS PS7. PSQ PSR8. PSQ PSR9. QSR is a straight 10. PSQ & PSR are supp.11. PSQ and PSR are rt s12. PS QR

1. Given2. Two points determine a seg.3. Radii of a circle are .4. Given5. A mdpt divides a segment into 2 segs.6. Reflexive property.7. SSS8. CPCTC9. Assumed from diagram.10.2 s that make a straight are supp. 11.If 2 s are both supp and , they are rt

s.12. If 2 lines intersect to form rt s, they

are .

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Τ

Given: ABCD is a rhombus

AB BC CD AD

Prove: AC BD

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Hint: Draw and label shape!

A

B C

D

1. AB BC CD AD2. AC AC3. BAC DAC4. 7 55. 3 46. ABE ADE7. 1 28. BED is a straight 9. 1 & 2 are supp.10. 1 and 2 are rt s11.AC BD

Τ

1. Given2. Reflexive Property3. SSS4. CPCTC5. If then6. ASA7. CPCTC8. Assumed from diagram.9. 2 s that make a straight are supp. 10.If 2 s are both supp and they are

rt s. 11.If 2 lines intersect and form rt s,

they are .Τ

1 E

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