Exercise 59 Page 422

Practice makes perfect

a Congruent triangles are identical. This means they have the same shape and size. Similar triangles, on the other hand, only have the same shape.

Are the Triangles Congruent?

Let's consider our triangles.

Congruent triangles have three pairs of congruent corresponding sides. Looking at our diagram, we can see that there are no congruent sides. Therefore, they cannot be congruent triangles.

Are the Triangles Similar?

Similar triangles have three pairs of congruent angles. However, we do not have any information about the angle measures of our triangles.

Instead, we have to determine if they are similar by analyzing the sides of the triangles. In similar triangles, the ratio of corresponding sides is constant. Let's identify the corresponding sides of the triangles.

Let's compare the ratios of the corresponding sides to determine if the triangles are similar. cccccc 6/2 & ? = & 12/4 & ? =& 15/5 & [1.2em] & & ⇕ & & & [0.7em] 3 & = & 3 & = & 3 & ✓ Since the ratio of each set of corresponding sides is equal, the triangles are similar by the Side-Side-Side (SSS) Similarity Theorem.

b Let's consider our pair of triangles.

We have been given two angles and a side length for both triangles. If we can show that they have at least two pairs of congruent angles, we can prove similarity. If the side between them is also congruent, we can prove congruence. We know two angles in each triangle, so let's find the third angle using the Triangle Angle Sum Theorem.

Equation	Missing Angle
112^(∘)+20^(∘)+m∠ a=180^(∘)	m∠ a=48^(∘)
48^(∘)+20^(∘)+m∠ b=180^(∘)	m∠ b=112^(∘)

As we can see, the triangles have congruent angles, so they are similar by the Angle-Angle (AA) Similarity Theorem.

Notice that the given length in the triangles is the included side to two pairs of congruent corresponding angles. From this we can prove congruence by the Angle-Side-Angle (ASA) Congruence Theorem

c We are given two right triangles.

We can use the Pythagorean Theorem to find the hypotenuse in the triangle on the right.

a^2+b^2=c^2

SubstituteII

a= 12, b= 5

12^2+ 5^2=c^2

▼

Solve for c

CalcPow

Calculate power

144+25=c^2

AddTerms

Add terms

169=c^2

RearrangeEqn

Rearrange equation

c^2=169

SqrtEqn

sqrt(LHS)=sqrt(RHS)

c=± 13

c > 0

c=13

Let's add the length of the hypotenuse to the triangle on the right.

Because the values of the hypotenuses and a pair of legs are the same, the triangles are congruent by the Hypotenuse-Leg Theorem.

d Let's consider the final pair of triangles.

Like we noted in Part B, if two triangles have at least two pairs of congruent angles, they are similar according to the AA Similarity Theorem. Since these triangles have two pairs of congruent angles, they are similar, but are they congruent? Let's identify corresponding sides.

For the triangles to be congruent, we would need to identify at least one pair of congruent corresponding sides. Even though the given sides are congruent, they are not corresponding because they are between different pairs of congruent angles.

However, if the triangles are isosceles with 48^(∘) base angles, the legs of each triangle would be congruent according to the Converse to the Base Angles Theorem. Let's use the Triangle Angle Sum Theorem again to find the missing angle ∠ θ. 62^(∘)+48^(∘)+m∠ θ = 180^(∘) ⇕ m∠ θ = 70^(∘) The triangles are not isosceles, so we cannot claim congruence.