Section 4.3-A Right Section 4.3-A Right Angle TheoremAngle TheoremBy Scott Salmon and J.C. By Scott Salmon and J.C.

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Theorem # 23Theorem # 23 If two angles are both If two angles are both

supplementary and supplementary and congruent, then they congruent, then they are right angles.are right angles.

WHAT????WHAT????

Since angle 1 and angle 2 form a straight Since angle 1 and angle 2 form a straight angle, they are supplementary. Therefore, angle, they are supplementary. Therefore, angle 1 plus angle 2 equals 180 degrees.angle 1 plus angle 2 equals 180 degrees.

Sample Problem#1Sample Problem#1

AB

C

Q

Given: CA congruent CB

AQ congruent QB

Prove: angle AQC and angle BQC are right angles

AnswerAnswer

Statements Reasons

1: CA congruent CB2: AQ congruent QB3: Angle AQC is suppl. to angle BQC 4: Angles AQC and BQC are right angles

1: Given

2: Given

3: If 2 adj. angles form a straight angle( Assumed from diagram), then the angles are suppl.

4: If 2 angles are both suppl. and congruent, then they are right angles

Sample Problem #2Sample Problem #2

Z W

S QTA

Given: Triangles SZT and AZT are congruent

Prove: Angles STZ and ATZ are right angles

AnswerAnswer

Statements Reasons

1: Triangles SZT and AZT are congruent2: Angle STZ is congruent to angle AZT3:Angle STZ is suppl. to angle AZT4:Angle STZ and Angle AZT are right angles

1: Given

2: CPCTC

3: IF two adj. angles form a straight angle, then they are suppl.

4: If two angles are both suppl. and congruent, then they are right angles

Practice Problem #1Practice Problem #1

A

B C

D

65

Given: Angle ADB is compl. to BDC andAngle ABDis congruent to angle DBC

Find : Measures of ALL angles

AnswerAnswer

A

D

BC

65

Explanation: Angle ADB is 35 degrees (subtraction), ABD is suppl. To CBD (Assumed from diagram)Therefore, angles ABD and CBD are right angles (theorem 23), Angle BCD is 35 degrees (subtraction) and angle BAD is 65 degrees (subtraction)

(35)

(90) (90) (35)(65)

Practice Problem #2Practice Problem #2

R

O

Y

P

TGiven: Angle TPYis congruent to Angle OPY,Angle TPY is suppl. To angle OPY

Prove: TYOR is a rhombus

AnswerAnswer

Statements Reasons

1: Angle TPYis congruent to Angle OPY2: Angle TPY is suppl. to Angle OPY3. Angle TPY and OPY are right angles4: Angle RPT is congruent to OPY, and Angle RPO is congruent to Angle YPT5:TYOR is a rhombus

1: Given

2: Given

3: Theorem 23 (a right angle theorem)

4: Vertical angles are congruent

5: If a quad’s diagonals are perpendicular, then the quad is a rhombus

Works CitedWorks Cited

Rhoad, Richard, George Milauskas, and Rhoad, Richard, George Milauskas, and Robert Whipple. Robert Whipple. Geometry for Geometry for

Enjoyment and ChallengeEnjoyment and Challenge. Evanston, . Evanston, IL: McDougal, Littell, 1991. Print.IL: McDougal, Littell, 1991. Print.

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