Properties of Triangles - SSAT Upper Level Quantitative (Math)

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Question

One angle of a right triangle has measure $68^{\circ }$ . Give the measures of the other two angles.

Answer

One of the angles of a right triangle is by definition a right, or $90^{\circ }$ , angle, so this is the measure of one of the missing angles. Since the measures of the angles of a triangle total $180^{\circ }$ , if we let the measure of the third angle be , then:

The other two angles measure $22 ^{\circ }, 90^{\circ }$ .

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Question

One angle of a right triangle has measure $120^{\circ }$ . Give the measures of the other two angles.

Answer

A right triangle must have one right angle and two acute angles; this means that no angle of a right triangle can be obtuse. But since $120^{\circ } > 90^{\circ }$ , it is obtuse. This makes it impossible for a right triangle to have a $120^{\circ }$ angle.

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Question

Find the degree measure of in the right triangle below.

Answer

The total number of degrees in a triangle is .

While $47^{\circ}$ is provided as the measure of one of the angles in the diagram, you are also told that the triangle is a right triangle, meaning that it must contain a $90^{\circ}$ angle as well. To find the value of , subtract the other two degree measures from .

$x=180-90-47=43^{\circ}$

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Question

Find the angle value of .

Answer

All the angles in a triangle must add up to 180 degrees.

$90^{\circ}+47^{\circ}+v=180^{\circ}$

$137^{\circ}+v=180^{\circ}$

$137^{\circ}+v-137^{\circ}=180^{\circ}-137^{\circ}$

$v=43^{\circ}$

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Question

Find the angle value of .

Answer

All the angles in a triangle adds up to $180^{\circ}$ .

$90^{\circ}+32^{\circ}+w=180^{\circ}$

$122^{\circ}+w=180^{\circ}$

$122^{\circ}+w-122^{\circ}=180^{\circ}-122^{\circ}$

$w=58^{\circ}$

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Question

Find the angle value of .

Answer

All the angles in a triangle add up to degrees.

$90^{\circ}+44^{\circ}+y=180^{\circ}$

$134^{\circ}+y=180^{\circ}$

$134^{\circ}+y-134^{\circ}=180^{\circ}-134^{\circ}$

$y=46^{\circ}$

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Question

Find the angle measure of .

Answer

All the angles in a triangle add up to $180^{\circ}$ .

$90^{\circ}+59^{\circ}+z=180^{\circ}$

$149^{\circ}+z=180^{\circ}$

$149^{\circ}+z-149^{\circ}=180^{\circ}-149^{\circ}$

$z=31^{\circ}$

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Question

If the vertex angle of an isoceles triangle is $64^{\circ}$ , what is the value of one of its base angles?

Answer

In an isosceles triangle, the base angles are the same. Also, the three angles of a triangle add up to $180^{\circ}$ .

So, subtract the vertex angle from $180^{\circ}$ . You get $116^{\circ}$ .

Because there are two base angles you divide $116^{\circ}$ by , and you get $58^{\circ}$ .

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Question

Figure NOT drawn to scale.

If $x = 72^{\circ }$ and $y = 43^{\circ }$ , evaluate .

Answer

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so

$w = x + y = 72 + 43 = 115 ^{\circ }$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram.

$45^{\circ } \leq m \angle 1 \leq 50^{\circ }$

$60^{\circ } \leq m \angle 2 \leq 70 ^{\circ }$

Which of the following could be a measure of $\angle 3$ ?

Answer

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so

$\angle 3 = \angle 1 + \angle 2$ .

We also have the following constraints:

$45^{\circ } \leq m \angle 1 \leq 50^{\circ }$

$60^{\circ } \leq m \angle 2 \leq 70 ^{\circ }$

Then, by the addition property of inequalities,

$45^{\circ } + 60 ^{\circ }\leq m \angle 1 +m \angle 2 \leq 50^{\circ } + 70 ^{\circ }$

$105 ^{\circ }\leq m \angle 3 \leq 120^{\circ }$

Therefore, the measure of $\angle 3$ must fall in that range. Of the given choices, only $110^{\circ }$ falls in that range.

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Question

Refer to the above diagram.

$45^{\circ } \leq m \angle 1 \leq 50^{\circ }$

$130^{\circ } \leq m \angle 3 \leq 140 ^{\circ }$

Which of the following could be a measure of $\angle 2$ ?

Answer

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so

$\angle 3 = \angle 1 + \angle 2$

or

$\angle 2 = \angle 3 - \angle 1$

Therefore, the maximum value of $\angle 2$ is the least possible value of $\angle 1$ subtracted from the greatest possible value of $\angle 3$ :

$140^{\circ } - 45^{\circ } = 95 ^{\circ }$

The minimum value of $\angle 2$ is the greatest possible value of $\angle 1$ subtracted from the least possible value of $\angle 3$ :

$130^{\circ } - 50^{\circ } = 80 ^{\circ }$

Therefore,

$80^{\circ } \leq \angle 2 \leq 95^{\circ }$

Since all of the choices fall in this range, all are possible measures of $\angle 2$ .

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Question

Find the angle measurement of .

Answer

All the angles in a triangle must add up to .

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Question

Find the angle measurement of .

Answer

All the angles in a triangle must add up to

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Question

Find the angle measurement of .

Answer

All the angles in a triangle must add up to .

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Question

The interior angles of a triangle measure $x^{\circ } , \left ( x+ 26\right )^{\circ } , \left (2x+ 1\right )^{\circ }$ . Of these three degree measures, give the greatest.

Answer

The degree measures of the interior angles of a triangle total 180 degrees, so

$\frac{4x}{4} = \frac{153}{4}$

$x = 38 \frac{1}{4}$

One angle measures $38\frac{1}{4}^{\circ }$

The other two angles measure

$x+ 26 = 38\frac{1}{4} + 26 = 64\frac{1}{4} ^{\circ }$

and

$2x+ 1 = 2 \left\(38\frac{1}{4}\right\)+ 1 = 77 \frac{1}{2} ^{\circ }$ .

We want the greatest of the three, or $77 \frac{1}{2} ^{\circ }$ .

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Question

An isosceles triangle has an angle whose measure is $70^{\circ }$ .

What could be the measures of one of its other angles?

(a) $40^{\circ }$

(b) $55^{\circ }$

(c) $70^{\circ }$

Answer

By the Isosceles Triangle Theorem, an isosceles triangle has two congruent interior angles. There are two possible scenarios if one angle has measure $70^{\circ }$ :

Scenario 1: The other two angles are congruent to each other. The degree measures of the interior angles of a triangle total $180^{\circ }$ , so if we let be the common measure of those angles:

This makes (b) a possible answer.

Scenario 2: One of the other angles measures $70^{\circ }$ also, making (c) a possible answer. The degree measure of the third angle is

,

making (a) a possible answer. Therefore, the correct choice is (a), (b), or (c).

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Question

One of the interior angles of a scalene triangle measures $54^{\circ }$ . Which of the following could be the measure of another of its interior angles?

Answer

A scalene triangle has three sides of different measure, so, by way of the Converse of the Isosceles Triangle Theorem, each angle is of different measure as well. We can therefore eliminate $54^{\circ }$ immediately.

Also, if the triangle also has a $72^{\circ }$ angle, then, since the total of the degree measures of the angles is $180^{\circ }$ , it follows that the third angle has measure

$180 - (54+72) = 180 - 126 = 54 ^{\circ }$ .

Therefore, the triangle has two angles that measure the same, and $72^{\circ }$ can be eliminated.

Similarly, if the triangle also has a $63^{\circ }$ angle, then, since the total of the degree measures of the angles is $180^{\circ }$ , it follows that the third angle has measure

$180 - (54+63) = 180 - 117= 63 ^{\circ }$ .

The triangle has two angles that measure $63^{\circ }$ . This choice can be eliminated.

$126^{\circ }$ can be eliminated, since the third angle would have measure

$180 - (54+126) = 180 - 180= 0^{\circ }$ ,

an impossible situation since angle measures must be positive.

The remaining possibility is $108^{\circ }$ . This would mean that the third angle has measure

$180 - (54+108) = 180 - 162= 18^{\circ }$ .

The three angles have different measures, so the triangle is scalene. $108^{\circ }$ is the correct choice.

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Question

Given: $\bigtriangleup ABC$ with $m \angle A = 20^{\circ }, m \angle B = 32^{\circ }$ . Locate on $\overline{AB}$ so that $\overrightarrow{CD}$ is the angle bisector of $\angle ACB$ . What is $m \angle CDB$ ?

Answer

Above is the figure described.

The measures of the interior angles of a triangle total $180^{\circ }$ , so the measure of $\angle ACB$ is

$m \angle ACB = 180^{\circ } -( m \angle A + m \angle B)$

$= 180^{\circ } -( 20^{\circ } + 32^{\circ })$

$= 180^{\circ } - 52^{\circ }$

$= 128^{\circ }$

Since $\overrightarrow{CD}$ bisects this angle,

$m \angle BCD= \frac{1}{2}m \angle ACB = \frac{1}{2} \cdot 128^{\circ } =64^{\circ }$

and

$m \angle CDB = 180^{\circ } -( m \angle BCD + m \angle B)$

$= 180^{\circ } -( 64 ^{\circ } + 32^{\circ } )$

$= 180^{\circ } -96^{\circ }$

$= 84^{\circ }$

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Question

Given: $\bigtriangleup ABC$ with $m \angle B = 74^{\circ }$ . is located on $\overline{AB}$ so that $\overrightarrow{CD}$ bisects $\angleACB$ $\angle ACB$ and forms isosceles triangle $\bigtriangleup BCD$ .

Give the measure of $\angle A$ .

Answer

If $\bigtriangleup BCD$ is isosceles, then by the Isosceles Triangle Theorem, two of its angles must be congruent.

Case 1: $m \angle B = m \angle BCD= 74^{\circ }$

Since $\overrightarrow{CD}$ bisects $\angleACB$ $\angle ACB$ into two congruent angles, one of which must be $m \angle BCD$ ,

$m \angle ACB = 2 \cdot m \angle BCD = 2 \cdot 74^{\circ } = 148^{\circ }$

However, this is impossible, since $\angle ACB$ and $\angle B$ are two angles of the original triangle; their total measure is

$m \angle ACB+ m \angle B = 148^{\circ }+ 74 ^{\circ } = 222 ^{\circ } > 180 ^{\circ }$

Case 2: $m \angle B = m \angle BDC= 74^{\circ }$

Then, since the degree measures of the interior angles of a triangle total $180^{\circ }$ ,

$m \angle BCD= 180^{\circ } - ( \angle B + m \angle BDC )$

$= 180^{\circ } - (74^{\circ } + 74^{\circ } )$

$= 180 ^{\circ } - 148^{\circ }$

$= 32^{\circ }$

Since $\overrightarrow{CD}$ bisects $\angleACB$ $\angle ACB$ into two congruent angles, one of which must be $m \angle BCD$ ,

$m \angle ACB = 2 \cdot m \angle BCD = 2 \cdot 32^{\circ } = 64^{\circ }$

and

$m \angle A = 180^{\circ } - ( \angle B + m \angle ACB )$

$= 180^{\circ } - (74^{\circ } + 64^{\circ } )$

$= 180^{\circ } - 138^{\circ }$

$= 42^{\circ }$

Case 3: $m \angle BDC= m \angle BCD$

Then

$m \angle B = 180^{\circ } -(m \angle BDC+ m \angle BCD)$

$74= 180^{\circ } -(m \angle BCD+ m \angle BCD)$

$74= 180^{\circ } -2 \cdot m \angle BCD$

$2 \cdot m \angle BCD = 106^{\circ }$

$m \angle BCD = 53^{\circ }$

$m \angle ACB = 2 \cdot m \angle BCD = 2 \cdot 53^{\circ } = 106^{\circ }$

$m \angle A = 180^{\circ } - ( \angle B + m \angle ACB )$

$= 180^{\circ } - ( 74^{\circ }+ 106^{\circ } )$

$= 0^{\circ }$ , which is not possible.

Therefore, the only possible measure of $\angle A$ is $42^{\circ }$ .

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Question

$\bigtriangleup ABC$ is a right triangle with right angle $\angle A$ . is located on $\overline{AB}$ so that, when $\overline{CD}$ is constructed, isosceles triangles $\bigtriangleup ADC$ and $\bigtriangleup BDC$ are formed.

What is the measure of $\angle B$ ?

Answer

The figure referenced is below:

Since $\bigtriangleup ADC$ is an isosceles right triangle, its acute angles - in particular, $\angle ADC$ - measure $45^{\circ }$ each. Since this angle forms a linear pair with $\angle CDB$ :

$m \angle CDB = 180^{\circ } - m \angle ADC = 180^{\circ } - 45 ^{\circ } = 135^{\circ }$ .

$\bigtriangleup BDC$ is also isosceles, so, by the Isosceles Triangle Theorem, it has two congruent angles. Since $\angle CDB$ is obtuse, and no triangle has two obtuse angles:

$m \angle BCD = m \angle B$ .

Also, $\angle ADC$ is an exterior angle of $\bigtriangleup BDC$ , whose measure is equal to the sum of those of its two remote interior angles, which are the congruent angles $m \angle BCD = m \angle B$ . Therefore,

$m \angle ADC = m \angle BCD + m \angle B$

$45^{\circ }= m \angle B + m \angle B$

$45^{\circ }=2 m \angle B$

$m \angle B = 22\frac{1}{2}^{\circ }$

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