Lesson 9

Conditions for Triangle Similarity

  • Let’s prove some triangles similar.

Problem 1

What is the length of segment \(DF\)?

Triangle A B C and D E F.

Problem 2

In triangle \(ABC\), angle \(A\) is 35º and angle \(B\) is 20º. Select all triangles which are similar to triangle \(ABC\).

A:

triangle \(DEF\) where angle \(D\) is 35º and angle \(E\) is 20º 

B:

triangle \(GHI\) where angle \(G\) is 35º and angle \(I\) is 30º 

C:

triangle \(JKL\) where angle \(J\) is 35º and angle \(L\) is 125º 

D:

triangle \(MNO\) where angle \(N\) is 20º and angle \(O\) is 125º 

E:

triangle \(PQR\) where angle \(Q\) is 20º and angle \(R\) is 30º

Problem 3

Decide whether triangles \(ABC\) and \(DEC\) are similar. Explain or show your reasoning.

Triangles A B C and D E C. Angle C A B labeled 36, angle A B C labeled 21. Angle C D E labeled 36, angle D C E labeled 123.

Problem 4

Lin is trying to convince Andre that all circles are similar. Help her write a valid justification for why all circles are similar. 

(From Unit 3, Lesson 8.)

Problem 5

Must these parallelograms be similar? Explain your reasoning. 

Quadrilateral A B C D. Length of A B is 2, A D is 6, B C is 6, and C D is 2.
Quadrilateral W X Y Z. W X, 3. X Y, 9. Y Z, 3. Z W, 9.  

​​​​​

(From Unit 3, Lesson 8.)

Problem 6

Determine if each statement must be true, could possibly be true, or definitely can't be true. Explain or show your reasoning.

  1. An equilateral triangle and a right triangle are similar.
  2. A right triangle and an isosceles triangle are similar.
(From Unit 3, Lesson 7.)

Problem 7

Quadrilaterals \(Q\) and \(P\) are similar.

What is the scale factor of the dilation that takes \(P\) to \(Q\)?

Quadrilaterals P and Q are similar. P has sides from the top measuring 4, 3 and 2, Q has sides from the top 5, none and 2 point 5. 
A:

\(\frac35\)

B:

\(\frac45\)

C:

\(\frac54\)

D:

\(\frac53\)

(From Unit 3, Lesson 6.)

Problem 8

The circle centered at \(Q\) is a scaled copy of the circle centered at \(R\). 

  1. Find the scale factor. 
  2. Find the value of \(x\).
A right triangle inscribed in a circle. Hypotenuse is labeled 5, and sides adjacent to the right angle are labeled 3 and x. Point R is on the hypotenuse.
A right triangle inscribed in a circle. Hypotenuse is labeled 20. Point Q is on the hypotenuse.
(From Unit 3, Lesson 1.)