Lesson 7

Angle-Side-Angle Triangle Congruence

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Problem 1

What triangle congruence theorem could you use to prove triangle \(ADE\) is congruent to triangle \(CBE\)?

Triangle ADE and CBE where angle DAE is congruent to and BCE. CE is congruent to AE.

Solution

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Problem 2

Han wrote a proof that triangle \(BCD\) is congruent to triangle \(DAB\). Han's proof is incomplete. How can Han fix his proof?

\(DC \parallel AB\)

Line DC is parallel to and above line AB and cut by transversal DB. Angles A and C are right angles.
  • Line \(AB\) is parallel to line \(DC\) and cut by transversal \(DB\). So angles \(CDB\) and \(ABD\) are alternate interior angles and must be congruent.
  • Side \(DB\) is congruent to side \(BD\) because they're the same segment.
  • Angle \(A\) is congruent to angle \(C\) because they're both right angles.
  • By the Angle-Side-Angle Triangle Congruence Theorem, triangle \(BCD\) is congruent to triangle \(DAB\).

Solution

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Problem 3

Segment \(GE\) is an angle bisector of both angle \(HEF\) and angle \(FGH\). Prove triangle \(HGE\) is congruent to triangle \(FGE\).

Quadrilateral E F G H. Segment G E is drawn in, dividing the shape into 2 triangles.

Solution

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Problem 4

Triangles \(ACD\) and \(BCD\) are isosceles. Angle \(BAC\) has a measure of 33 degrees and angle \(BDC\) has a measure of 35 degrees. Find the measure of angle \(ABD\).

Triangle ACD with point B near the center. Segments BD, BC, and BA are drawn. Line segment BD is congruent to BC.

Solution

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(From Unit 2, Lesson 6.)

Problem 5

Which conjecture is possible to prove?

A:

All triangles with at least one side length of 5 are congruent.

B:

All pentagons with at least one side length of 5 are congruent.

C:

All rectangles with at least one side length of 5 are congruent.

D:

All squares with at least one side length of 5 are congruent.

Solution

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(From Unit 2, Lesson 5.)

Problem 6

Andre is drawing a triangle that is congruent to this one. He begins by constructing an angle congruent to angle \(LKJ\). What is the least amount of additional information that Andre needs to construct a triangle congruent to this one?

Triangle JKL.

Solution

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(From Unit 2, Lesson 4.)

Problem 7

Here is a diagram of a straightedge and compass construction. \(C\) is the center of one circle, and \(B\) is the center of the other. Which segment has the same length as segment \(CA\)?

 
Two circles, one large centered at point C. Points A, B, and D lie on the large circle. The smaller circle centered at point B intersects larger circle at points A and D. Radii CA, CB, and CD are drawn. Radii BA and BD are drawn.
 
A:

\(BA\)

B:

\(BD\)

C:

\(CB\)

D:

\(AD\)

Solution

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(From Unit 1, Lesson 1.)