Similar Triangles

We will use the Angle-Angle Similarity Theorem to determine if the given triangles are similar. Let's first take a look at the given triangles.

We can see that ∠ A and ∠ D are congruent since they both measure 70^(∘). However, ∠ C and angle ∠ E are not congruent. That being said, ∠ B might still be congruent with ∠ E. We can determine the measure of ∠ B by using the Interior Angles Theorem, which states that the sum of all interior angles is 180^(∘). Interior Angles Theorem m∠ A+ m∠ B + m∠ C=180^(∘) Therefore, we can subtract m∠ A and m∠ C from 180^(∘) to find m∠ B. m∠ B=180^(∘)-70^(∘)-39^(∘) ⇓ m∠ B=71^(∘) This means that m∠ B is 71^(∘).

Now that we know m∠ B, we can see that ∠ B and ∠ E are congruent. Since we have two pair of congruent angles, the Angle-Angle Similarity Theorem tells us that the triangles are similar. ∠ A ≅ ∠ D ∠ B ≅ ∠ E ⇓ △ ABC ~ △ DEF

We will use a similar process to determine whether the triangles are similar. Start by examining the given triangles.

We have two angles, ∠ A and ∠ D, that are congruent. Let's see if we can find another pair of congruent angles by determining the measure of ∠ B. To do so, let's subtract m∠ A and m∠ C from 180^(∘). m∠ B=180^(∘) - 63^(∘) -55^(∘) ⇓ m ∠ B = 62^(∘) We can find m∠ F using a similar process. m∠ F=180^(∘) - 55^(∘) -55^(∘) ⇓ m ∠ F = 70^(∘) Therefore, m∠ B is 62^(∘) and m∠ F is 70^(∘).

Now that we know all the angle measures, we can see that there are no two pairs of congruent angles between these triangles. Therefore, we can conclude that the triangles are not similar.

Let's apply a similar fashion to find out if the following triangles are similar.

We can observe that ∠ C and ∠ F are congruent since they have the same measure. Next, we will check if we can identify another pair of congruent angles. To do so, we have to calculate m∠ A by subtracting m∠ B and m∠ C from 180^(∘). m∠ A=180^(∘) - 95^(∘) -47^(∘) ⇓ m ∠ A = 38^(∘) The measure of angle A is 38^(∘).

From the diagram, we can see that ∠ A is congruent to ∠ D. We have now found two pairs of congruent angles between △ ABC and △ DEF. This means that the two triangles are similar. ∠ A ≅ ∠ D ∠ C ≅ ∠ F ⇓ △ ABC ~ △ DEF

We are told that △ ABC and △ DEF are similar triangles.

Because of the similarity, the ratio between corresponding sides is equal. In this case, AB corresponds to DE and AC corresponds to DF. We can use this information to write a proportion that relates the side lengths of the two triangles. AB/DE=AC/DF We have been given the values of AB, AC, and DE, so we can substitute these values into the proportion and solve for DF.

This means that DF equals 4.8 feet.

This time, we need to determine the value of EF. Let's begin by considering the given triangles.

Since the triangles are similar, we know that the ratio between corresponding sides is equal. In this case, DE corresponds to AB and EF corresponds to BC. Therefore, we can set up the following proportion. DE/AB=EF/BC We can substitute the values of DE, AB, and BC, and solve the proportion for EF.

Therefore, EF is 6 centimeters.

Let's start by looking at the given diagram.

Assuming that the tree and the dog are perpendicular to the ground, they create right angles with the ground. In addition, the angles formed by the rays of the Sun when they hit each object and the ground are equal. This implies that there are two pairs of congruent angles, resulting in two similar triangles as per the Angle-Angle Similarity Theorem.

This means the ratio of the tree's height to the dog's height is equal to the ratio of the tree's shadow to the dog's shadow since the corresponding sides of similar triangles are proportional. Tree's Height/Dog's Height=Tree's Shadow/Dog's Shadow ⇓ h/2=25/5 Let's solve this proportion to find the value of h, which corresponds to the height of the tree.

Therefore, the height of the tree is 10 feet.

Let's begin by considering the given diagram.

Let's assume that the house and the school are positioned at right angles to the ground. Also, the angles created by the Sun's rays as they hit each object and the ground are equal. This results in two pairs of congruent angles, which satisfies the Angle-Angle Similarity Theorem and creates two similar triangles.

This means that the ratio of the school's height to the house's height equals the ratio of the school's shadow to the house's shadow. This is because the ratio between corresponding sides of similar triangles are equal. School's Height/House's Height=School's Shadow/House's Shadow ⇓ h/3=12/6 Let's solve this proportion to find the value of h.

Therefore, the height of the school is 6 meters.