Question

Which group contains triangles that are all similar?

Answer 1

Answer:

Triangles represented by option B are similar triangles.

Step-by-step explanation:

Please find the attachment.

We have been given 4 group of triangles and we are asked to find which group contains similar triangles.

A. We can see from our diagram that triangles provided in option A are not similar.

The first triangle is an isosceles right triangle as it has 2 angles each having measure 45 degrees. By angle sum property the measure of 3rd angle will be:

$180^o-(45^o+45^o)=180^o-90^o=90^o$

The second triangle is an equilateral triangle as all the sides equal to 7 units, so all angles will be equal to 60 degrees.

The 3rd triangle is an isosceles triangle as  it has 2 angles each having measure 53 degrees. By angle sum property the measure of 3rd angle will be:

$180^o-(53^o+53^o)=180^o-106^o=74^o$

We can see that none of the triangles have same angle measures, therefore, option A is not a correct choice.

B. The triangles provided in option B are similar triangles as each triangle have one right angle and they are sharing the same vertex.

Let us assume that the common vertex has x degrees angle, then by angle sum property the measure of 3rd angle in each triangle will be $180-90-x=90-x$ degrees. Therefore, the triangles provided in option B are similar triangles by AAA similarity theorem.

C. The triangles provided in option C are not similar triangles as we are not told that the given lines are parallel or not. In addition the bottom figure is a quadrilateral as it has 4 angles, Therefore, the given triangles are not similar triangles.

D. We can see that our given triangles have vertical angles, so their measure will be 32 degrees. Since the given side lengths are different, therefore, this group of triangles is not similar.

Answer 2

Option B is correct as the triangles are similar.

Further Explanation:

The similar triangles are those in which all the corresponding angles are equal and the sides are proportional.

There are many similarity rules and are as follows.

1. Angle Angle (AA)

2. Side SideSide (SSS)

3. Side Angle Side (SAS)

Explanation:

In option (a)

The two angles of the first triangle are ${45^ \circ }{\text{ and }}{45^ \circ }.$

The third angle can be obtained as follows,

$\begin{aligned}{\text{Angle}}&= {180^ \circ } - {45^ \circ } - {45^ \circ }\\&= {90^ \circ }\\\end{aligned}$

The second triangle is equilateral. Therefore, all the angles are of ${60^ \circ }.$

The two angles of the third triangle are ${53^ \circ }{\text{ and }}{53^ \circ }.$

The third angle can be obtained as follows,

$\begin{aligned}{\text{Angle}}&= {180^ \circ } - {53^ \circ } - {53^ \circ }\\&= {74^ \circ }\\\end{aligned}$

The triangles are not similar.

In option (b)

All the triangles in the figure are similar to each other as the one angle is of ${90^ \circ }$ and the other angle is common.

Therefore, the triangles are similar to each other by AA Similarity Theorem.

In option (c)

The triangles are not similar to each other.

In option (d)

The triangles are not similar to each other.

Option B is correct as the triangles are similar.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Triangle

Keywords: similar, two triangles, congruent, angles, triangle, ASA, angle side angle, congruent sides, acute angle, side, corresponding angles, congruent triangle, similarity theorem, SSS congruency theorem, SSS similarity theorem, AA similarity postulate, SAS congruency postulate.