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HSPT Math : Concepts

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Example Questions

Example Question #108 : Properties Of Triangles

Triangle

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 $\angle 3$ ?

Possible Answers:

$110^{\circ }$

$125^{\circ }$

$130^{\circ }$

All of the other choices give a possible measure of $\angle 3$ .

$100^{\circ }$

Correct answer:

$110^{\circ }$

Explanation:

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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Example Question #109 : Properties Of Triangles

Triangle

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 $\angle 2$ ?

Possible Answers:

$85 ^{\circ }$

$90 ^{\circ }$

$95 ^{\circ }$

$80 ^{\circ }$

All of the other responses are correct.

Correct answer:

All of the other responses are correct.

Explanation:

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

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

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

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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Example Question #1411 : Concepts

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

Possible Answers:

This triangle cannot exist.

$30^{\circ }, 30^{\circ }$

$30^{\circ }, 90^{\circ }$

$120^{\circ }, 120^{\circ }$

$90^{\circ }, 120^{\circ }$

Correct answer:

This triangle cannot exist.

Explanation:

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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Example Question #2 : How To Find An Angle In A Right Triangle

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

Possible Answers:

$52^{\circ }, 60^{\circ }$

$44^{\circ },68^{\circ }$

$22^{\circ }, 90^{\circ }$

$56^{\circ },56^{\circ }$

This triangle cannot exist.

Correct answer:

$22^{\circ }, 90^{\circ }$

Explanation:

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:

x + 68 + 90 = 180

x + 158 = 180

x + 158 - 158= 180 - 158

x = 22

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

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Example Question #1 : How To Find An Angle Of A Line

Varsity_question

AB and CD are two parrellel lines intersected by line EF. If the measure of angle 1 is 132^o , what is the measure of angle 2?

Possible Answers:

$48^{\circ}$

$24^{\circ}$

$64^{\circ}$

$180^{\circ}$

$132^{\circ}$

Correct answer:

$132^{\circ}$

Explanation:

The angles are equal. When two parallel lines are intersected by a transversal, the corresponding angles have the same measure.

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Example Question #2 : How To Find An Angle Of A Line

Lines A and B in the diagram below are parallel. The triangle at the bottom of the figure is an isosceles triangle.

Act2

What is the degree measure of angle ?

Possible Answers:

Correct answer:

Explanation:

Since A and B are parallel, and the triangle is isosceles, we can use the supplementary rule for the two angles, and 2x+30 which will sum up to 180 . Setting up an algebraic equation for this, we get 4x+2x+30=180 . Solving for , we get x=25 . With this, we can get either 2(25)+30=80 (for the smaller angle) or 4(25)=100 (for the larger angle - must then use supplementary rule again for inner smaller angle). Either way, we find that the inner angles at the top are 80 degrees each. Since the sum of the angles within a triangle must equal 180, we can set up the equation as 80+80+y=180

y=20 degrees.

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Example Question #1 : How To Find An Angle In A Trapezoid

Find the measure of angle in the isosceles trapezoid pictured below.

Screen_shot_2013-03-18_at_3.32.40_pm

Possible Answers:

$108^{\circ}$

$18^{\circ}$

$118^{\circ}$

$72^{\circ}$

$36^{\circ}$

Correct answer:

$108^{\circ}$

Explanation:

The sum of the angles in any quadrilateral is 360°, and the properties of an isosceles trapezoid dictate that the sets of angles adjoined by parallel lines (in this case, the bottom set and top set of angles) are equal. Subtracting 2(72°) from 360° gives the sum of the two top angles, and dividing the resulting 216° by 2 yields the measurement of x, which is 108°.

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Example Question #2 : Acute / Obtuse Triangles

Two interior angles in an obtuse triangle measure $123^{\circ}$ and $11^{\circ}$ . What is the measurement of the third angle.

Possible Answers:

$123^{\circ}$

$46^{\circ}$

$104^{\circ}$

$50^{\circ}$

$57^{\circ}$

Correct answer:

$46^{\circ}$

Explanation:

Interior angles of a triangle always add up to 180 degrees.

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Example Question #1 : Acute / Obtuse Triangles

In the triangle below, AB=BC (figure is not to scale) . If angle A is 41°, what is the measure of angle B?

A (Angle A = 41°)

Act_math_108_02

B C

Possible Answers:

Correct answer:

Explanation:

If angle A is 41°, then angle C must also be 41°, since AB=BC. So, the sum of these 2 angles is:

41° + 41° = 82°

Since the sum of the angles in a triangle is 180°, you can find out the measure of the remaining angle by subtracting 82 from 180:

180° - 82° = 98°

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Example Question #21 : Acute / Obtuse Triangles

Points A, B, C, D are collinear. The measure of ∠ DCE is 130° and of ∠ AEC is 80°. Find the measure of ∠ EAD.

Screen_shot_2013-03-18_at_3.27.08_pm

Possible Answers:

60°

80°

70°

50°

Correct answer:

50°

Explanation:

To solve this question, you need to remember that the sum of the angles in a triangle is 180°. You also need to remember supplementary angles. If you know what ∠ DCE is, you also know what ∠ ECA is. Hence you know two angles of the triangle, 180°-80°-50°= 50°.

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All HSPT Math Resources

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HSPT Math : Concepts

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #108 : Properties Of Triangles

Example Question #109 : Properties Of Triangles

Example Question #1411 : Concepts

Example Question #2 : How To Find An Angle In A Right Triangle

Example Question #1 : How To Find An Angle Of A Line

Example Question #2 : How To Find An Angle Of A Line

Example Question #1 : How To Find An Angle In A Trapezoid

Example Question #2 : Acute / Obtuse Triangles

Example Question #1 : Acute / Obtuse Triangles

Example Question #21 : Acute / Obtuse Triangles

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