8.05 Proving triangles similar

Now that we can recognize similar triangles, we now demonstrate how to formally prove that two triangles are similar. These proofs will be a variation on the kinds of proofs we encountered when demonstrating congruence.

Before we get started, note that we write "the measures of two pairs of sides are in the same proportion" using fractions. So in a pair of triangles like these:

... if we are given $$ABJL​=ACKL​=BCKJ​ we know that the proportion of $$AB to $$JL is the same as the proportion of $$AC to $$KL, which is also the same as the proportion of $$BC to $$KJ. This allows us to conclude that $$ΔABC~ΔJKL, by $$SSS~.

Having established similarity, we can then solve for missing values, like we did for congruence.

 

Worked examples

Question 1

In the following triangles, $$QZSZ​=PZRZ​. Prove that these triangles are similar.

Think: We have been given two side measures in the same proportion, and the angles in between are vertical angles. We should use $$SAS~ to prove these triangles are similar.

Do: We write down what we want to prove, followed by the information obtained from the information given to us:

To prove: $$ΔPQZ~ΔRSZ
Statement	Reason
$$QZSZ​=PZRZ​	Given
$$∠PZQ≅∠RZS	Vertical angles are congruent
$$ΔPQZ~ΔRSZ	$$SAS~

Question 2

Prove that $$ΔWXZ is similar to $$ΔXYZ, and solve for the value of $$t:

Think: We have been given that there are two pairs of congruent angles, so we can demonstrate that these triangles are similar using AA. Once that is done, we can use the fact that these triangles are similar to solve for $$t after matching up the sides correctly.

Do: We write down what we want to find, followed by the information obtained from the markings and the measures given on the diagram:

To find: the value of $$t
Statement	Reason
$$∠WZX≅∠XZY	Given
$$∠WXZ≅∠XYZ	Given
$$ΔWXZ~ΔXYZ	$$AA~
$$ZW=1	Given
$$ZX=3	Given
$$ZY=t	Given
$$ZYZX​=ZXZW​	Corresponding sides of similar triangles are in the same proportion
$$t3​=31​	Substitution
$$t=9	Multiplication property of equality

question 3

Prove that $$ΔCGD and $$ΔEDF are similar, and solve for the value of $$θ:

Think: Here we are only given one angle measure that doesn't involve a variable, so we will need to rely on the sides to demonstrate similarity. Once that is done, we can match up the angles and form an equation to solve for $$θ.

Do: We write down what we want to find, followed by the information obtained from the markings and the measures on the diagram:

To find: the value of $$θ
Statement	Reason
$$CG=4	Given
$$ED=2	Given
$$CGED​=4ED​	Multiplication property of equality
$$CGED​=2	Substitution
$$CD=5	Given
$$EF=2.5	Given
$$CDEF​=5EF​	Multiplication property of equality
$$CDEF​=2	Substitution
$$GD=GF+FD	Segment addition theorem
$$GF=FD	Given
$$GD=2FD	Substitution
$$GDFD​=2	Multiplication property of equality
$$CGED​=CDEF​=GDFD​	Transitive property of equality
$$ΔCGD~ΔEDF	$$SSS~
$$∠CGD=∠EDF	Corresponding angles of similar triangles are congruent
$$m∠CGD=m∠EDF	Congruent angles have equal measure
$$m∠CGD=54°	Given
$$m∠EDF=5θ−6°	Given
$$54=5θ−6	Substitution
$$60=5θ	Addition property of equality
$$5θ=60	Symmetric property of equality
$$θ=12	Multiplication property of equality

 

Practice questions

question 4

question 5

question 6