How to find if two acute / obtuse triangles are similar - Intermediate Geometry

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Question

Are the two triangles similar? If so, which postulate proves their similarity? (Figure not drawn to scale)

Answer

The triangles are similar by the SSS postulate. The proportions of corresponding sides are all equal.

$\frac{3}{9}=\frac{1}{3}$

$\frac{4}{12}=\frac{1}{3}$

$\frac{6}{18}=\frac{1}{3}$

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Question

Are the two triangles similar? If so, which postulate proves their similarity? (Figure not drawn to scale)

Answer

The triangles are similar by the angle-angle postulate. 2 corresponding angles are equal to each other, therefore, the triangles must be similar.

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Question

Are the two triangles similar? If so, which postulate proves their similarity? (Figure not drawn to scale)

Answer

The triangles are not similar, and it can be proven through the side-angle-side postulate. The SAS postulate states that two sides flanking a corresponding angle must be similar. In this case, the angles are congruent. However, the sides are not similar.

$\frac{12}{3}=4$

$\frac{18}{4}=4.5$

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Question

If the two triangles shown above are similar, what is the measurements for angles and ?

Answer

In order for two triangles to be similar, they must have equivalent interior angles.

Thus, angle degrees and angle degrees.

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Question

Using the similar triangles above, find a possible measurement for sides and .

Answer

Since the two triangles are similar, each triangles three corresponding sides must have the same ratio.

The original ratio of side lengths is:

Thus a similar triangle will have this same ratio:

$3:8\rightarrow 3\cdot 2:8\cdot2\rightarrow 6:16$

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Question

Triangle one and triangle two are similar triangles. Triangle one has two sides with lengths and . What are possible measurements for the corresponding sides in triangle two?

Answer

Since the two triangles are similar, each triangles three corresponding sides must have the same ratio.

The ratio of side lengths for triangle one is:

Thus the ratio of side lengths for the second triangle must following this as well:

$5:7\rightarrow 5\cdot 3: 7\cdot 3=15:21$ , because both side lengths in triangle one have been multiplied by a factor of .

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Question

Triangle one and triangle two are similar triangles. Triangle one has two sides with lengths and . What are possible measurements for the corresponding sides in triangle two?

Answer

Since the two triangles are similar, each triangles three corresponding sides must have the same ratio.

The ratio of triangle one is:

$6:2\rightarrow 3:1$

Therefore, looking at the possible solutions we see that one answer has the same ratio as triangle one.

$3\cdot 8: 1\cdot 8 \rightarrow 24:8=3:1$

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Question

Triangle one and triangle two are similar triangles. Triangle one has two sides with lengths and . What are possible measurements for the corresponding sides in triangle two?

Answer

Since the two triangles are similar, each triangles three corresponding sides must have the same ratio.

The ratio of the side lengths in triangle one is:

If we take this ratio and look at the possible solutions we will see:

$16:9=(16\times 2):(9\times 2)$

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Question

Triangle one and triangle two are similar triangles. Triangle one has two sides with lengths mm and mm. What are possible measurements for the corresponding sides in triangle two?

Answer

Since the two triangles are similar, each triangles three corresponding sides must have the same ratio.

The ratio of triangle one is:

If we look at the possible solutions we will see that ratio that is in triangle one is also seen in the triangle with side lengths as follows:

$23:35=(23\times 4):(35\times 4)=92:140$

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Question

Using the triangle shown above, find possible measurements for the corresponding sides of a similar triangle?

Answer

Since the two triangles are similar, each triangles three corresponding sides must have the same ratio.

The ratio of the triangle is:

Applying this ratio we are able to find the lengths of a similar triangle.

$7:16=(7\times 3):(16\times3)=21:48$

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Question

Using the triangle shown above, find possible measurements for the corresponding sides of a similar triangle?

Answer

Since the two triangles are similar, each triangles three corresponding sides must have the same ratio.

The ratio of the triangle is:

Applying this ratio we are able to find the lengths of a similar triangle.

$29:33=(29\times 5):(33\times 5)=145:165$

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$m\angle B = 78 ^{\circ }$

$m\angle E = 78 ^{\circ }$

True or false: It follows from the given information that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

Answer

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

According to the Side-Angle-Side Similarity Theorem (SASS), if the lengths of two pairs of corresponding sides of two triangles are in proportion, and their included angles are congruent, then the triangles are similar.

$\overline{AB}$ and $\overline{DE}$ are corresponding sides, as are $\overline{BC}$ and $\overline{EF}$ ; $\angle B$ and $\angle E$ are their included angles. Substituting

$\frac{AB}{DE} = \frac{10}{12}= \frac{5}{6}$

$\frac{BC}{EF} = \frac{15}{18} = \frac{5}{6}$

Therefore, $\frac{AB}{DE} =\frac{BC}{EF}$ , and corresponding sides are in proportion.

$m \angle B = 78^{\circ }$ and $m \angle E = 78^{\circ }$ ; the included angles are congruent.

The conditions of SASS are met, and it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

True or false: From the above six statements, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

Answer

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

If $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then corresponding sides must be in proportion; that is, it must hold that

$\frac{AB}{DE} = \frac{AC}{DF} = \frac{BC}{EF}$

Substituting the lengths of the sides for the respective quantities:

$\frac{AB}{DE} = \frac{12}{18} = \frac{2}{3}$

$\frac{AC}{DF} = \frac{18}{30} = \frac{3}{5}$

The inequality of these two side ratios disproves the similarity of the triangles, so the correct answer is "false".

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$m \angle A = 56 ^{\circ }$

$m \angle B = 72 ^{\circ }$

$m \angle E = 72^{\circ }$

$m \angle F = 52^{\circ }$

True or false: From the above four statements, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

Answer

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

The sum of the measures of the interior angles of a triangle is $180^{\circ }$ . Therefore,

$m \angle A + m \angle B + m \angle C = 180^{\circ }$

Set $m \angle A = 56 ^{\circ }$ and $m \angle B = 72 ^{\circ }$ , and solve for $m \angle C$ :

$56 ^{\circ } + 72 ^{\circ } + m \angle C = 180^{\circ }$

$128 ^{\circ } + m \angle C = 180^{\circ }$

$128 ^{\circ } + m \angle C - 128 ^{\circ } = 180^{\circ } - 128 ^{\circ }$

$m \angle C = 52 ^{\circ }$

By the Angle-Angle Similarity Postulate (AA), two triangles are similar if two angles of the first triangle are congruent to those of their counterparts in the second. $m \angle B = 72 ^{\circ }$ and $m \angle E = 72^{\circ }$ , so $\angle B \cong \angle E$ ; $m \angle C = 52 ^{\circ }$ and $m \angle F = 52^{\circ }$ , so $\angle C \cong \angle F$ . The conditions of AA are met, so it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ . The correct response is "true".

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\angle A \cong \angle C$

$\angle B \cong \angle E$

$\angle D \cong \angle F$

True or false: From the above three statements, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

Answer

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

Let $x = m \angle B = m \angle E$ .

Let $y = m \angle A = m \angle C$

The measures of the interior angles of a triangle total $180^{\circ }$ . Therefore,

$m \angle A + m \angle B + m \angle C = 180^{\circ }$

Substituting:

$y + x + y= 180^{\circ }$

$2 y + x = 180^{\circ }$

$2 y + x - x = 180^{\circ }- x$

$2 y = 180^{\circ }- x$

$\frac{2y}{2} = \frac{180^{\circ }- x}{2}$

$y = \frac{180^{\circ }- x}{2}$

$m \angle A = \frac{180^{\circ }- x}{2}$

By the same reasoning,

$m \angle D = \frac{180^{\circ }- x}{2}$

Therefore, $\angle A \cong \angle D$ .

By the Angle-Angle Similarity Postulate (AA), two triangles are similar if two angles of the first triangle are congruent to those of their counterparts in the second. $\angle A \cong \angle D$ and $\angle B \cong \angle E$ , so the conditions of AA are met; it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

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Question

Given: $\bigtriangleup ABC$ , with Point on $\overline{AB}$ and Point on $\overline{AC}$ .

True or false: From the given information, it follows that $\bigtriangleup ABC \sim \bigtriangleup AMN$ .

Answer

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup AMN$ , then , , and correspond respectively to , , and .

If two triangles are similar, then it must hold that corresponding sides are in proportion. Specifically, if $\bigtriangleup ABC \sim \bigtriangleup AMN$ , it must hold that

$\frac{AB}{AM} = \frac{AC}{AN}$

By the Segment Addition Postulate,

and

Set , , , in the above proportion statement, which becomes

$\frac{25}{10} = \frac{24}{8}$

Reduce both ratios to lowest terms:

$\frac{5}{2} = \frac{3}{1}$

The corresponding sides are not proportional, so the statement $\bigtriangleup ABC \sim \bigtriangleup AMN$ is false.

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Question

Given: $\bigtriangleup ABC$ , with Point on $\overline{AB}$ and Point on $\overline{AC}$ .

True or false: From the given information, it follows that $\bigtriangleup ABC \sim \bigtriangleup AMN$ .

Answer

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup AMN$ , then , , and correspond respectively to , , and .

By the Side-Angle-Side Similarity Theorem (SASS), two triangles are similar if two pairs of corresponding sides are in proportion and their included angles are congruent.

Examine two pairs of corresponding sides: $\overline{AB}$ and $\overline{AM}$ , and $\overline{AC}$ and $\overline{AN}$ . In both cases, their included angle is $\angle A$ ; by the Reflexive Property of Congruence, $\angle A \cong \angle A$ .

It remains to be demonstrated that

$\frac{AB}{AM} = \frac{AC}{AN}$

By the Segment Addition Postulate,

and

Set , , , in the above proportion statement, which becomes

$\frac{18}{6} = \frac{27}{9}$

Reduce both ratios to lowest terms:

$\frac{3}{1} = \frac{3}{1}$

The corresponding sides are proportional.

The conditions of SASS have been proved, so the statement $\bigtriangleup ABC \sim \bigtriangleup AMN$ is true.

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Question

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that

$\angle A \cong \angle D$

$\angle B \cong \angle E$

Which statement(s) must be true?

(a) $\bigtriangleup ABC \sim \bigtriangleup DEF$

(b) $\bigtriangleup ABC \cong \bigtriangleup DEF$

Answer

The two given angle congruences set up the conditions of the Angle-Angle Similarity Postulate - if two angles of one triangle are congruent to the two corresponding angles of another triangle, the two triangles are similar. It follows that

$\bigtriangleup ABC \sim \bigtriangleup DEF$ .

However, congruence cannot be proved, since at least one side congruence is needed to prove this. This is not given in the problem.

Therefore, statement (a) must hold, but not necessarily statement (b).

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