a Similar triangles have three pairs of congruent corresponding angles.
B
b In similar shapes, the ratio between any pair of corresponding sides is always the same.
C
c Do you have enough information from the two triangles to determine if they are similar?
A
a Similar
Explanation: See solution.
B
b Not similar
Explanation: See solution.
C
c Not enough information.
Explanation: See solution.
Practice makes perfect
a To begin with, we will label the two unknown angles a and b.
If two triangles are similar, they have three pairs of congruent corresponding angles. We already know one pair. To determine if the remaining angles are congruent, we need to find the third angle's measure. Using the Triangle Angle Sum Theorem, we can write and solve two equations.
&a+27^(∘)+47^(∘) =180^(∘)
&b+27^(∘)+106^(∘)=180^(∘)
Let's solve the first equation.
Since the triangles have three pairs of congruent corresponding angles, we know that the triangles are similar.
b In similar shapes, the ratio between any pair of corresponding sides is always equal. Let's identify corresponding sides in the two figures.
Again, if these figures are similar, the ratio between corresponding sides is the same.
98/18? =64/12
By calculating the two ratios, we can determine if the shapes are similar.
Since the ratios are not equal, the figures cannot be similar.
c The only information we have about the triangles are two angles in the smaller triangle and one angle in the larger triangle. Knowing two of the angles in the smaller triangle, we can by the Triangle Angle Sum Theorem, calculate the unknown angle which we can label a.
With this, we know that one pair of angles in the triangles are congruent. However, to decide if the triangles are similar, we need to know that at least two pairs of angles are congruent. We do not have enough information to decide if they are similar. Note that this does not mean they are not similar. We cannot say one way or the other.
