Triangle Similarity Theorems Application

Which triangle similarity theorem is used to prove that ΔGSP~ΔBSP?

• AA Similarity Theorem

• SAS Similarity Theorem

• SSS Similarity Theorem

• Right Triangle Similarity Theorem

Answer explanation

Angle S in ΔGSP is congruent to angle S in ΔBSP and angle P in ΔGSP is congruent to angle P in ΔBSP, the appropriate similarity theorem to prove ΔGSP~ΔBSP is the Angle-Angle (AA) Similarity Theorem.

The AA Similarity Theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. In this case, the angles S and P in ΔGSP are congruent to the corresponding angles S and P in ΔBSP, satisfying the conditions for the AA Similarity Theorem.

Apply SAS Similarity Theorem. What is the ratio of the given corresponding sides?

• 2

• SSS similarity theorem

• SAS similarity theorem

• SS similarity theorem

• AA similarity theorem 

Find the x and y, such that ΔSAY ~ Δ TWO.

• 4 and 10

• 4 and 12

• 2 and 12

• 2 and 10

• 8

• 9

• 10

• 11

A right triangle measures 8.5 cm, 14.72cm, and 17 cm each side.

What kind of special right triangle it is? Why?

• A 45-45-90 Special Right Triangle because the shorter leg is half the hypotenuse.

• A 30-60-90 Special Right Triangle because the longer leg is the product of  and the      shorter leg.

• A 30-60-90 Special Right Triangle because the shorter leg is half the hypotenuse and the longer leg is the product of the shorter leg and .

• It is not a special right triangle.

Answer explanation

To determine the type of special right triangle based on the given side lengths, we can analyze the ratios of the sides.

The given side lengths are:

• Side A: 8.5 cm

• Side B: 14.72 cm

• Hypotenuse (side C): 17 cm

To identify the special right triangle, we can compare the ratios of the side lengths:

1. Checking for a Pythagorean triple:

• If the ratios of the side lengths form a Pythagorean triple (a set of three positive integers that satisfy the Pythagorean theorem), then it is a Pythagorean triple right triangle.

• A Pythagorean triple right triangle has side lengths that are integer multiples of a common ratio.

2. Checking for a 45-45-90 triangle:

• If the ratio of the shorter sides is 1:1, and the hypotenuse is √2 times the length of the shorter side, then it is a 45-45-90 triangle.

3. Checking for a 30-60-90 triangle:

• If the ratio of the shorter side to the longer side is 1:√3, and the ratio of the shorter side to the hypotenuse is 1:2, then it is a 30-60-90 triangle.

Now, let's calculate the ratios:

Ratio of side A to side B: 8.5 cm / 14.72 cm ≈ 0.577 Ratio of side B to side C: 14.72 cm / 17 cm ≈ 0.866

Based on the ratios, we see that they do not match any of the ratios for a Pythagorean triple, 45-45-90, or 30-60-90 triangle.

Therefore, the given triangle with side lengths 8.5 cm, 14.72 cm, and 17 cm is not a special right triangle. It is a general right triangle with no specific special properties in terms of its side ratios.

ΔBIG is a right triangle with a shorter leg 8cm and a longer leg 17cm.

What is the perimeter of ΔBIG? Round to the nearest hundredth?

• 42.79 cm

• 43.79 cm

• 47.392 cm

• 43.29 cm

John climbed up on a Mango Tree. He was 5 meters above the ground when he decided to climbed down but he suddenly stopped and asked his father for help. His father brought a 6m ladder, leaned it on the mango tree and help John to get down.

How far is the foot of the ladder from the base of the Mango Tree?

• 3.32 m

• 4 m

• 4.32 m

• 3.92 m

• Distributive property

• Reflexive property

• Symmetric property

• Transitive property

Answer explanation

The property of congruence that supports the statement that the two triangles D and G are congruent by the AA (Angle-Angle) similarity theorem is the Reflexive Property of Congruence.

The Reflexive Property of Congruence states that any geometric figure is congruent to itself. In this case, the congruence statement D ≅ G implies that triangle D is congruent to triangle G. Since the AA similarity theorem requires that the corresponding angles of the two triangles are congruent, we can say that triangle G is congruent to itself, and therefore, triangle G ≅ G.

Using the Reflexive Property of Congruence, we can establish that triangle G is congruent to triangle D in order to support the conclusion made using the AA similarity theorem.

Which of the following figures does NOT illustrate similarity?

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