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HSPT Math : Geometry

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Example Questions

Example Question #281 : Geometry

$\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$m \angle A = m \angle C = 60 ^{\circ }$ .

Is $\bigtriangleup DEF$ scalene, isosceles, or equilateral - or can it be determined?

Possible Answers:

$\bigtriangleup DEF$ is a isosceles triangle, but not equilateral.

$\bigtriangleup DEF$ is an equilateral triangle.

Whether $\bigtriangleup DEF$ is scalene, isosceles, or equilateral cannot be determined.

$\bigtriangleup DEF$ is a scalene triangle.

Correct answer:

$\bigtriangleup DEF$ is an equilateral triangle.

Explanation:

Corresponding angles of similar triangles are congruent, so

$m \angle D = m \angle A = 60^{\circ }$

$m \angle F = m \angle C = 60^{\circ }$

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

$m \angle E =180^{\circ } - (m \angle D +m \angle F )$

$m \angle E =180^{\circ } - (60^{\circ } +60^{\circ } )$

$m \angle E =60^{\circ }$

The three angles of $\bigtriangleup DEF$ all measure the same, so it is equiangular; consequently, it is also equilateral.

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Example Question #282 : Geometry

In parallelogram ABCD , $m \angle A = (t+ 56 )^{\circ }$ . Give the measure of $\angle C$ in terms of .

Possible Answers:

$(t+ 146)^{\circ }$

$(124-t )^{\circ }$

$(t+ 56 )^{\circ }$

$(34-t )^{\circ }$

Correct answer:

$(t+ 56 )^{\circ }$

Explanation:

$\angle A$ and $\angle C$ are a pair of opposite angles of the parallelogram, and as such, they are congruent. Therefore,

$m \angle C = m \angle A = (t+ 56 )^{\circ }$ .

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Example Question #1467 : Concepts

The measures of the angles of $\bigtriangleup ABC$ are as follows:

$m \angle A = (t+35)^{\circ }$

$m \angle B = (3t-15)^{\circ }$

$m \angle C =(85-t)^{\circ }$

Is $\bigtriangleup ABC$ scalene, isosceles, or equilateral?

Possible Answers:

$\bigtriangleup ABC$ cannot exist

$\bigtriangleup ABC$ is equilateral

$\bigtriangleup ABC$ is isosceles but not equilateral

$\bigtriangleup ABC$ is scalene

Correct answer:

$\bigtriangleup ABC$ is equilateral

Explanation:

The sum of the measures of the angles of a triangle is 180 degrees, so solve for in the equation:

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

$(t+35)^{\circ }+ (3t-15)^{\circ } + (85-t)^{\circ }= 180 ^{\circ }$

t+3t-t+35-15+85 = 180

3t +105= 180

3t +105- 105 = 180 - 105

3t =75

$3t \div 3 =75 \div 3$

t = 25

$m \angle A = (t+35)^{\circ } = (25+35)^{\circ } =60^{\circ }$

$m \angle B = (3t-15)^{\circ }= (3 \cdot 25-15)^{\circ } = (75-15)^{\circ } = 60^{\circ }$

$m \angle C =(85-t)^{\circ } =(85-25)^{\circ } = 60^{\circ}$

A triangle with three 60-degree angles is equilateral.

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Example Question #283 : Geometry

$\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$m \angle A = m \angle C = 45 ^{\circ }$ .

Is $\bigtriangleup DEF$ acute, right, or obtuse - or can it be determined?

Possible Answers:

$\bigtriangleup DEF$ is an acute triangle.

$\bigtriangleup DEF$ is a right triangle.

Whether $\bigtriangleup DEF$ is acute, right, or obtuse cannot be determined.

$\bigtriangleup DEF$ is an obtuse triangle.

Correct answer:

$\bigtriangleup DEF$ is a right triangle.

Explanation:

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

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

$m \angle B =180^{\circ } - (45 ^{\circ } +45 ^{\circ } )$

$m \angle B =90^{\circ }$

Corresponding angles of similar triangles are congruent, so

$m \angle E = m \angle B = 90 ^{\circ }$

$\angle E$ is right, so $\bigtriangleup DEF$ is a right triangle.

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Example Question #1469 : Concepts

$m \angle A = 40^{\circ }$ in isosceles triangle $\bigtriangleup ABC$ . Is this triangle acute, right, or obtuse?

Possible Answers:

$\bigtriangleup ABC$ is acute

It is inconclusive whether $\bigtriangleup ABC$ is acute, right, or obtuse

$\bigtriangleup ABC$ is right

$\bigtriangleup ABC$ is obtuse

Correct answer:

It is inconclusive whether $\bigtriangleup ABC$ is acute, right, or obtuse

Explanation:

An isosceles triangle has at least two sides of equal measure, and also has at least two angles of equal measure.

One of two things can hold:

$m \angle A = 40^{\circ }$ , so either of $\angle B$ or $\angle C$ has measure $40^{\circ }$ . The third angle has measure

$[180 - (40+40 ) ]^{\circ }= (180-80) ^{\circ } = 100^{\circ }$ .

This angle, having measure greater than 90 degrees, is obtuse, so $\bigtriangleup ABC$ an obtuse triangle.

Alternatively, $m \angle B = \angle C = t ^{\circ }$ for some , so

40 + t + t = 40 + 2t = 180

40 + 2t - 40 = 180 - 40

2t = 140

$2t \div 2 = 140 \div 2$

t = 70

$m \angle B = m \angle C = 70 ^{\circ }$

Since all three angles have measure less than 90 degrees, all three angles are acute, and $\bigtriangleup ABC$ is an acute triangle.

Therefore, the information is inconclusive.

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Example Question #284 : Geometry

In parallelogram ABCD , $m \angle A = (t+ 56 )^{\circ }$ , and $m \angle B = 2t^{\circ }$ . Evaluate .

Possible Answers:

t = 62

t = 72

t= 56

$t = 41 \frac{1}{3}$

Correct answer:

$t = 41 \frac{1}{3}$

Explanation:

$\angle A$ and $\angle B$ are a pair of adjacent angles of the parallelogram, and as such, they are supplementary - that is, their degree measures total 180. Therefore,

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

$(t+ 56)^{\circ } + 2t ^{\circ }= 180 ^{\circ }$

3t+56 = 180

3t+56 - 56 = 180 - 56

3t =124

$3t\div 3 =124 \div 3$

$t = 41 \frac{1}{3}$

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Example Question #281 : Geometry

What is the total degrees of the angles in a square?

Possible Answers:

270

180

360

Correct answer:

360

Explanation:

A square has four right angles which each have degrees.

To get the total you just multiple the measure of one by to get

90*4=360 .

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Example Question #1591 : Basic Geometry

Angle $\angle ABC$ measures $20^{\circ}$

$\overrightarrow{BD}$ is the bisector of $\angle ABC$

$\overrightarrow{BE}$ is the bisector of $\angle CBD$

What is the measure of $\angle ABE$ ?

Possible Answers:

$15^{\circ}$

$10^{\circ}$

$5^{\circ}$

$30^{\circ}$

$40^{\circ}$

Correct answer:

$15^{\circ}$

Explanation:

Angle pic

Let's begin by observing the larger angle. $\angle ABC$ is cut into two 10-degree angles by $\overrightarrow{BD}$ . This means that angles $\angle ABD$ and $\angle CBD$ equal 10 degrees. Next, we are told that $\overrightarrow{BE}$ bisects $\angle CBD$ , which creates two 5-degree angles. $\angle ABE$ consists of $\angle ABD$ , which is 10 degrees, and $\angle DBE$ , which is 5 degrees. We need to add the two angles together to solve the problem.

$\angle ABE=\angle ABD+\angle DBE$

$\angle ABE=10^{\circ}+5^{\circ}$

$\angle ABE=15^{\circ}$

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Example Question #11 : How To Find An Angle Of A Line

If $\overleftrightarrow{FH} \parallel \overleftrightarrow{AD}$ , $\overline{BE}\perp \overleftrightarrow{GC}$ , and $m\angle HGC=58^\circ$ , what is the measure, in degrees, of $\angle ABE$ ?

Alternate interior angles

Possible Answers:

148

122

Correct answer:

148

Explanation:

The question states that $\overleftrightarrow{FH} \parallel \overleftrightarrow{AD}$ . The alternate interior angle theorem states that if two parallel lines are cut by a transversal, then pairs of alternate interior angles are congruent; therefore, we know the following measure:

$m\angle GCA = 58^\circ$

The sum of angles of a triangle is equal to 180 degrees. The question states that $\overline{BE}\perp \overleftrightarrow{GC}$ ; therefore we know the following measure:

$m \angle BEC = 90^\circ$

Use this information to solve for the missing angle: $\angle EBC$

$180^\circ=m\angle EBC+58^\circ+90^\circ$

$m\angle EBC=32^\circ$

The degree measure of a straight line is 180 degrees; therefore, we can write the following equation:

$180^\circ=m\angle ABE+32^\circ$

$m\angle ABE=148^\circ$

The measure of $\angle ABE$ is 148 degrees.

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Example Question #282 : Geometry

What is the slope of a line through the points (2,5) and (-2,3) ?

Possible Answers:

$-\frac{1}{2}$

$\frac{1}{2}$

Undefined slope

Correct answer:

$\frac{1}{2}$

Explanation:

Use the slope fomula, setting $x_{1} = -2, x_{2} = 2,y_{1} = 3, y_{2} = 5$

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{5-3}{2-(-2)} = \frac{2}{4} =\frac{1}{2}$

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HSPT Math : Geometry

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #281 : Geometry

Example Question #282 : Geometry

Example Question #1467 : Concepts

Example Question #283 : Geometry

Example Question #1469 : Concepts

Example Question #284 : Geometry

Example Question #281 : Geometry

Example Question #1591 : Basic Geometry

Example Question #11 : How To Find An Angle Of A Line

Example Question #282 : Geometry

All HSPT Math Resources

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