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Common Core: High School - Geometry : High School: Geometry

Study concepts, example questions & explanations for Common Core: High School - Geometry

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All Common Core: High School - Geometry Resources

6 Diagnostic Tests 114 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #5 : Prove Theorems About Triangles.

True or False: The Base Angles Converse states that if two base angles are congruent, then their opposite sides are congruent.

Possible Answers:

False

True

Correct answer:

True

Explanation:

Since the Base Angles Theorem states that if two adjacent sides are congruent then their opposite angles are congruent, then its converse assumes that the two base angles are congruent and then their opposite sides are congruent.

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Example Question #3 : Prove Theorems About Triangles.

Consider the following theorem: If two triangles have two pairs of congruent corresponding sides and their included angle is congruent, then these two triangles are congruent. When proving this theorem, what information do we assume to begin the proof?

Possible Answers:

Assume that two triangles have two pairs of corresponding congruent sides

Assume that the two triangles are congruent

Assume that two triangles have a pair of corresponding congruent angles

Assume that two triangles have two pairs of corresponding congruent sides and their included angle is congruent

Correct answer:

Assume that two triangles have two pairs of corresponding congruent sides and their included angle is congruent

Explanation:

When theorems are presented to you in an “if-then” format, you always assume the entire “if” part. This allows you to construct an argument that you will be able to prove to be true using the given information.

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Example Question #7 : Prove Theorems About Triangles.

Fill in the blanks below to complete the proof for the HL Theorem. Use the figure below to find the given information.

Screen shot 2020 08 12 at 9.37.09 am

Screen shot 2020 08 12 at 9.37.44 am

Possible Answers:

perpendicular, altitude, $\angle DCA \cong \angle BCD$

altitude, perpendicular, $\angle DCA\cong \angle BCD$

angle bisector, bisector, $\overline{AD}\cong \overline{BD}$

Correct answer:

altitude, perpendicular, $\angle DCA\cong \angle BCD$

Explanation:

Follow the detailed proof below.

Screen shot 2020 08 12 at 9.38.29 am

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Example Question #8 : Prove Theorems About Triangles.

The triangle inequality theorem states that if you have a triangle ABC then

Fill in the blanks for the proof of the theorem below using triangle ABC

Screen shot 2020 08 12 at 9.53.30 am

To begin this proof we first must let there exist the triangle DAC , where the length of $\overline{DC}\cong \overline{AC}$ and $\overline{AC}$ is a shared side between the two triangles.

Screen shot 2020 08 12 at 9.55.00 am

Screen shot 2020 08 12 at 9.55.48 am

Possible Answers:

obtuse triangle, $\angle DAC< \angle DAB$

isosceles triangle, $\angle DAC< \angle DAB$

equilateral triangle, $\overline{AC}< \overline{CB}$

Correct answer:

isosceles triangle, $\angle DAC< \angle DAB$

Explanation:

Follow the detailed proof below for an explanation.

To begin this proof we first must let there exist the triangle DAC , where the length of $\overline{DC}\cong \overline{AC}$ and $\overline{AC}$ is a shared side between the two triangles.

Screen shot 2020 08 12 at 9.55.00 am

Screen shot 2020 08 12 at 9.58.20 am

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Example Question #3 : Prove Theorems About Triangles.

Fill in the blanks to the proof below. We are proving that the base angles of isosceles triangles, ABC , are congruent. This is a proof of the isosceles triangle theorem.

Screen shot 2020 08 13 at 8.12.44 am

Possible Answers:

$\overline{AB} \cong \overline{BA}$ , similar, angle bisector, SAS

$\overline{AC}\cong \overline{BC}$ , congruent, midpoint, ASA

$\overline{AC}\cong \overline{BC}$ , congruent, angle bisector, SAS

Correct answer:

$\overline{AC}\cong \overline{BC}$ , congruent, angle bisector, SAS

Explanation:

Screen shot 2020 08 13 at 8.14.18 am

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Example Question #10 : Prove Theorems About Triangles.

Prove that if one side of a triangle is longer than another side of the same triangle, then the angle opposite the longer side will be greater than the angle opposite the shorter side. Use the information that $\overline{AB }< \overline{AC}$ for triangle ABC .

Screen shot 2020 08 20 at 11.01.12 am

Possible Answers:

Proof:

Screen shot 2020 08 20 at 11.02.53 am

Proof:

Screen shot 2020 08 20 at 11.02.29 am

Proof:

Screen shot 2020 08 20 at 11.03.23 am

Correct answer:

Proof:

Screen shot 2020 08 20 at 11.02.53 am

Explanation:

Follow the detailed proof below.

Screen shot 2020 08 20 at 11.01.48 am

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Example Question #1 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Using the sides of a right triangle, what is the definition of $tan(\theta)$ ?

Possible Answers:

$tan(\theta)=\frac{adjacent}{hypotenuse}$

$tan(\theta)=\frac{opposite}{adjacent}$

$tan(\theta)=\frac{opposite}{hypotenuse}$

Correct answer:

$tan(\theta)=\frac{opposite}{adjacent}$

Explanation:

We can see this to be true by looking at the following triangle.

We are considering $\angle C$ . The opposite side to $\angle C$ is $\bar{AB}$ . $\angle C$ has two adjacent sides, but since $\bar{AC}$ is opposite to the right angle, this is the hypotenuse of the triangle. So the adjacent side to $\angle C$ must be $\bar{BC}$ . We can set up the following equation to check and make sure that $tan(\theta)=\frac{opposite}{adjacent}$ .

$tan(\theta)=\frac{opposite}{adjacent}$

$tan(36.87) = \frac{3}{4}$

0.75 = 0.75

This shows that $tan(\theta)=\frac{opposite}{adjacent}$ .

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Example Question #1 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

True or False: By similarity, side ratios in right triangles are properties of the angles in the triangle.

Possible Answers:

False

True

Correct answer:

True

Explanation:

Similar right triangles are two right triangles that differ in side lengths but have congruent corresponding angles. This means that if you have an angle, $\theta_1$ , in the first triangle and an angle, $\theta_2$ , in the second triangle. So $\theta_1 = \theta_2$ .If we are considering the cosine of these two angles.

$cos(\theta_1) = \cos(\theta_2)$

$\frac{adjacent_1}{hypotenuse_1} = \frac{adjacent_2}{hypotenuse_2}$

Side ratios would also follow from computing the sine and tangent of the angles using their sides as well. This shows that the side ratios are properties of the angles in the triangles.

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Example Question #3 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Assume the two triangles below are similar. Using the fact that their sides ratios are $\frac{BC}{EF} = \frac{AC}{DF}$ , what trigonometric function could this represent for angles and

Screen shot 2020 08 07 at 2.50.44 pm

Possible Answers:

cos(C) , cos(F)

sin(C) , cos(F)

cos(C) , sin(F)

sin(C) , sin(F)

Correct answer:

cos(C) , cos(F)

Explanation:

We must begin by manipulating our equation of the side ratios to get fractions that include both of the sides of the same triangles:

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

$\implies BC = \frac{EF*AC}{DF}$ multiply both sides by

$\implies \frac{BC}{AC} = \frac{EF}{DF}$ divide both sides by

If we look at angle we see that $\frac{BC}{AC}$ is the $\frac{adjacent}{hypotenuse}$ . Looking at angle we see that EFDF is also $\frac{adjacent}{hypotenuse}$ . This is the definition of the cosine function of an angle. So:

$cos(C) = \frac{BC}{AC} = \frac{EF}{DF} = cos(F)$

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Example Question #4 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Consider the following right triangle. Use trigonometric ratios to solve for and .

Screen shot 2020 08 12 at 9.41.24 am

Possible Answers:

$AB = \sqrt{3}, AC = 2$

$AB = 1, AC = \sqrt{3}$

AB = 3, AC = 1

AB = 1, AC = 2

Correct answer:

AB = 1, AC = 2

Explanation:

We are given angle and we need to find sides and . So first we need to think about what relation these sides are to angle . Side is the hypotenuse of the triangle since it is the opposite of the right angle. Side is opposite angle . Recall that the trigonometric ratio that corresponds to sine is $\frac{opposite}{hypotenuse}$ . We can solve for the missing sides by solving for the following equation.

$sin(C) = \frac{opposite}{hypotenuse}$

$sin(30) = \frac{AB}{AC}$

$\frac{1}{2} = \frac{AB}{AC}$

It follows that:

1 = AB

2 = AC

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All Common Core: High School - Geometry Resources

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Common Core: High School - Geometry : High School: Geometry

Study concepts, example questions & explanations for Common Core: High School - Geometry

All Common Core: High School - Geometry Resources

Example Questions

Example Question #5 : Prove Theorems About Triangles.

Example Question #3 : Prove Theorems About Triangles.

Example Question #7 : Prove Theorems About Triangles.

Example Question #8 : Prove Theorems About Triangles.

Example Question #3 : Prove Theorems About Triangles.

Example Question #10 : Prove Theorems About Triangles.

Example Question #1 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Example Question #1 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Example Question #3 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

Example Question #4 : Understand That By Similarity, Side Ratios In Right Triangles Are Properties Of The Angles In The Triangle, Leading To Definitions Of Trigonometric Ratios For Acute Angles

All Common Core: High School - Geometry Resources

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