apex algebra with trig and stats learning packet · 2020-04-10 · ll theorem again, the triangles...

Algebra with Trig and Stats: Weeks 3 and 4, April 20 – May 1

CHARLES COUNTY PUBLIC SCHOOLS

APEX Algebra with Trig and Stats

Learning Packet


Student: _________________________________ School: _____________________________

Teacher: _________________________________ Block/Period: ________________________

Packet Directions for Students NOTE: This assignment covers two weeks and is due on May 1

Read through the Instruction and examples on Congruent Right Triangles while completing the corresponding questions on the 6.2.1 Study Guide

Complete 6.2.1 Study Guide

o Check and revise solutions using the 6.2.1 Study Guide Answer Key

Complete Quiz: Proving Right Triangle Congruence

Complete Quiz: Right Triangle Measurements


Congruent Right Triangles

The shortcuts are coming! The shortcuts are coming! Do you remember all those shortcut names for triangle congruence postulates? SSS, SAS, ASA, AAS Well get ready for some more. Only this time, the postulates are only for right triangles. And this time, they're going to save you even more time. Your shortcuts are about to get shorter! Objectives

Explore the right triangle congruence shortcut theorems — HL, LL, HA, and LA. Discover the relationship between right triangle congruence theorems and the

congruence theorems for non-right triangles. Use the perpendicular bisector theorem to find unknown side lengths or determine if

two right triangles are congruent. Use the angle bisector theorem to find unknown angle measures in right triangles.


In this lesson, you will learn about congruent right triangles.

Here is what you already know about congruent triangles:

They have the same size and same shape.

All of their corresponding side lengths and angle measures are equal.

Right Triangles

Now, what if two congruent triangles are right triangles?

Congruent right triangles have the same properties that congruent nonright triangles have —

same size and shape, three pairs of congruent sides, and three pairs of congruent angles.


Congruence Shortcuts

Here is a listing of the congruence postulates and theorems that can be used to show that two

triangles are congruent.

Congruence Postulates and Theorems

These Do Not Work

Can these be simplified for right triangles? After all, two right triangles automatically have one

pair of congruent angles — the right angles.

The Hypotenuse and One Leg

The first question is, can you build two congruent right triangles if the measures for only

the hypotenuse and one leg are known?


HL Theorem

You may have seen that your triangles were always congruent. This brings us to the first right

triangle congruence shortcut, the HL congruence theorem. "HL" stands for "hypotenuse-leg."

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and

corresponding leg of another right triangle, then the right triangles are congruent.

Given: A and D are right angles.

AB DE

BC EF

Conclusion: ABC DEF

Reason: HL

LL Theorem

Again, the triangles were always congruent. You just discovered the LL congruence theorem.

"LL" stands for "leg-leg."

If the legs of one right triangle are congruent to the corresponding legs of another right triangle,

then the right triangles are congruent.

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Given: I and L are right angles.

chatTranscript

GI JL

HI KL

Conclusion: GHI JKL

Reason: LL


A Special Case of SSS

Why does the HL congruence theorem guarantee congruence?

Recall that when you know two side lengths in a right triangle, you can calculate the third with

the Pythagorean theorem. So, if two pairs of sides are congruent, the third pair must also be

congruent.

Using the same logic, you can conclude that the LL congruence theorem also

guarantees congruence. Since LL means two pairs of legs are congruent, then, by

the Pythagorean theorem, the hypotenuse of the triangles must be congruent.

In addition, the angles included between the two pairs of congruent legs are right angles and are

also congruent.So, LL is a special case of both the SSS postulate and the SAS postulate!

You can use the LL theorem to prove that c = f without using the Pythagorean theorem.

1. The LL theorem proves that the two triangles are congruent.

2. CPCTC proves that c = f.

Confirm:

Theorem: HL Theorem

How can you use the hypotenuse-leg theorem to test for congruence? If the hypotenuse and a leg

of one right triangle are congruent to the hypotenuse and corresponding leg of another right

triangle, then the triangles are congruent.

Theorem: LL Theorem

How can you use the leg-leg theorem to test for congruence? [Click.] If the legs of one right

triangle are congruent to the corresponding legs of another right triangle, then the triangles are

congruent.

HA Theorem

The HA congruence theorem states that if the hypotenuse and an acute angle of one right triangle

are congruent to the hypotenuse and corresponding acute angle of another right triangle, then

the right triangles are congruent.

"HA" stands for "hypotenuse-angle."

Given: M and P are right angles.

O R

NO QR


Conclusion: MNO PQR

Reason: HA

One Leg and an Included Angle

Finally, is "leg-angle" a right triangle congruence shortcut?

Note that there are two possible legs to consider. One leg is included between the right angle and

the other given angle, and one leg is not.

What about the other "leg-angle" case, in which the leg is not included between the given angles?

Does this guarantee congruent triangles?

LA Theorem

You may have noticed that both cases of "leg-angle" guarantee congruence. This is the fourth

and final right triangle congruence theorem, the LA congruence theorem.

If the leg and an acute angle of one right triangle are congruent to the leg and corresponding

acute angle of another right triangle, then the triangles are congruent.

Confirm:

Theorem: HA Theorem

How can you use the hypotenuse-angle theorem to test for congruence? If the hypotenuse and an

acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle

of another right triangle, then the triangles are congruent.

Theorem: LA Theorem

How can you use the leg-angle theorem to test for congruence? If the leg and an acute angle of

one right triangle are congruent to the leg and corresponding acute angle of another right

triangle, then the triangles are congruent.


Right Triangle Congruence Theorems

You have now discovered four congruence theorems for use with right triangles. They are

summarized in the table below.

Notice that each is really just a special case of an earlier congruence theorem.

Theorems Using Congruent Right Triangles

Congruent right triangles are the key to proving several important theorems:

Perpendicular bisector theorem

Perpendicular bisector theorem converse

Angle bisector theorem

Angle bisector theorem converse

Another key to understanding these theorems is to remember that the distance from a point to a

line is always measured along the perpendicular segment that joins them.

Perpendicular Bisector Theorem

Mathematical practice

The perpendicular bisector theorem says that if a point is on the perpendicular bisector of a

segment, then it is equidistant from the endpoints of the segmen


Converse of the Perpendicular Bisector Theorem

The converse of the perpendicular bisector theorem is also true.

If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector

of that segment.

Angle Bisector Theorem

Just as a segment can be bisected by a line, so can an angle.

The angle bisector theorem states that if a point is on the bisector of an angle, then it is

equidistant from the two sides of the angle.

Converse of the Angle Bisector Theorem

The converse of the angle bisector theorem is true as well.

If a point is in the interior of an angle and is equidistant from the angle's two sides, then it lies on

the bisector of the angle.


6.2.1 Study: Congruent Right Triangles

Use the questions below to keep track of key concepts from this lesson's study activity.

1) Practice: Activating Prior Knowledge and Summarizing

Fill in the blanks in the list.


These triangles have the exact same and .

Their pairs of corresponding sides are congruent.

Their three pairs of corresponding are congruent.

2) Practice: Using Visual Cues

Complete the congruence statements for the two right triangles below.

Circle the statement that names the congruent right angles.

Congruent Sides Congruent Angles


3) Practice: Organizing Information

Complete the chart.

Congruence Shortcuts for Right Triangles

Theorem Description

HL theorem

(hypotenuse-

leg)

If the and of a right triangle are congruent

to the and corresponding of another right

triangle, then the two triangles are congruent.

LL theorem

(leg-leg)

If the of a right triangle are congruent to

the of another right triangle, then the two triangles are

congruent.

HA theorem

(hypotenuse-

angle)

If the and an angle of a right triangle are

congruent to the and corresponding angle

of another right triangle, then the two triangles are congruent.

LA theorem

(leg-angle)


congruent to the corresponding and angle


4) Practice: Asking Questions

Fill in the blanks.

Why is each right triangle congruence shortcut a special case of a triangle congruence shortcut?

1. All right triangles have a given pair of congruent angles — their angles.

2. If you know the length of two sides of a right triangle, you can use

the theorem to find the third side. So, if two pairs of sides are congruent,

the third pair is too!


5) Practice: Organizing Information

Complete the chart.

Theorem Special Case of

HL theorem (hypotenuse-leg) postulate

LL theorem (leg-leg) postulate

and postulate

HA theorem (hypotenuse-angle) theorem

LA theorem (leg-angle) theorem

and postulate


Label each pair of right triangles with the theorem that proves they are congruent.


7) Practice: Summarizing and Using Visual Cues Fill in the blank in the rule. Then use the rule to solve each problem.

Rule: The distance from a point to a line is the segment that joins them.

What is the distance from C to line AB?


8) Practice: Drawing Inferences and Summarizing

Fill in the blanks.


Use the four theorems above to solve each problem.

What is the distance from C to B?

What is the measure of DCR?


6.2.1 Study: Congruent Right Triangles

ANSWER KEY

1) Practice: Activating Prior Knowledge and Summarizing (Pages 1 – 2)

Fill in the blanks in the list.


These triangles have the exact same and .

size; shape

Their pairs of corresponding sides are congruent.

three

Their three pairs of corresponding are congruent.

angles

2) Practice: Using Visual Cues (Page 2)

Complete the congruence statements for the two right triangles below.

Circle the statement that names the congruent right angles.

Congruent Sides Congruent Angles


3) Practice: Organizing Information (Pages 4 – 16)

Complete the chart.

Congruence Shortcuts for Right Triangles

Theorem Description

HL theorem

(hypotenuse-

leg)

If the and of a right triangle are congruent

to the and corresponding of another right

triangle, then the two triangles are congruent.

hypotenuse; leg; hypotenuse; leg (Page 5)

LL theorem

(leg-leg)

If the of a right triangle are congruent to

the of another right triangle, then the two triangles are

congruent.

legs; legs; (Page 7)

HA theorem

(hypotenuse-

angle)


congruent to the and corresponding angle


hypotenuse; acute; hypotenuse; acute (Page 13)

LA theorem

(leg-angle)


congruent to the corresponding and angle


leg; acute; leg; acute (Page 16)

4) Practice: Asking Questions (Pages 8 – 9 and 17)

Fill in the blanks.

Why is each right triangle congruence shortcut a special case of a triangle congruence shortcut?

1. All right triangles have a given pair of congruent angles — their angles.

right

2. If you know the length of two sides of a right triangle, you can use

the theorem to find the third side. So, if two pairs of sides are congruent,

the third pair is too!

Pythagorean


5) Practice: Organizing Information (Pages 8 – 9 and 17)

Complete the chart.

Theorem Special Case of

HL theorem (hypotenuse-leg) postulate

SSS (side-side-side)

LL theorem (leg-leg) postulate and postulate

SSS (side-side-side); SAS (side-angle-side)

HA theorem (hypotenuse-angle) theorem

AAS (angle-angle-side)

LA theorem (leg-angle) theorem and postulate

AAS (angle-angle-side); ASA (angle-side-angle)

6) Practice: Using Visual Cues (Pages 11, 19 – 21)

Label each pair of right triangles with the theorem that proves they are congruent.

LA theorem

HL theorem

LL theorem

HA theorem


7) Practice: Summarizing and Using Visual Cues (Page 22)

Fill in the blank in the rule.

Then use the rule to solve each problem.

Rule: The distance from a point to a line is the segment that joins them.

perpendicular



4

1.5

8) Practice: Drawing Inferences and Summarizing (Pages 23 – 30)

Fill in the blanks.


9) Practice: Using Visual Cues (Pages 23 – 30)

Use the four theorems above to solve each problem.

What is the distance from C to B?

What is the measure of DCR?

21

31°


Quiz: Proving Right Triangle Congruence Question 1a of 10

Based on the information marked in the diagram, LMO and FGH must be congruent.

# Choice

A. True

B. False

Question 2a of 10

Based on the information marked in the diagram, XYZ and RST must be congruent.

# Choice

A. True

B. False


Question 3a of 10

Based only on the information given in the diagram, which congruence theorems or postulates

could be given as reasons why JKL QRS? Check all that apply.

B. HL

C. LL

D. HA

E. LA

F. SAS


Question 4a of 10


could be given as reasons why CDE OPQ? Check all that apply.

A. AAS

B. ASA

C. LL

D. SAS

E. HL

F. LA


Question 5a of 10


could be given as reasons why ABC UVW?

Check all that apply.

# Choice

A. LL

B. LA

C. HL

D. ASA

E. SAS


Question 6a of 10


could be given as reasons why DEF KLM?


# Choice

A. LA

B. AAS

C. HA

D. SAS

E. LL

F. HL


Question 7a of 10


could be given as reasons why HIJ KLM?


# Choice

A. ASA

B. LL

C. LA

D. HL

E. HA

F. AAS


Question 8a of 10

Which of the following are right triangle congruence theorems?


A. Hypotenuse-leg (HL)

B. Hypotenuse-angle (HA)

C. Leg-angle (LA)

D. Leg-leg (LL)

Question 9a of 10

The HL theorem is a special case of the _____.

A. SSS postulate

B. ASA postulate

C. SAS postulate

D. AAS theorem

Question 10a of 10

The LA theorem is a special case of the _____.

A. SAS postulate and ASA postulate

B. SAS postulate and SSS postulate

C. AAS theorem and ASA postulate

D. AAS theorem and SSS postulate


Quiz: Right Triangle Measurements Question 1a of 10

What is the distance from point N to in the figure below?

# Choice

A. 7.6

B. 9.7

C. 0.8

D. 6.9

E. 2.9

F. 6.8


Question 2a of 10

What is the distance from point Y to in the figure below?

# Choice

A. 10

B. 35

C. Cannot be determined

D.

E. 5

F.


Question 3a of 10

What is the measure of RCP in the figure below?

# Choice

A. 35

B. 55

C. 11

D. 60

E. 70

F. Cannot be determined


Question 4a of 10

What is the measure of ABC in the figure below?

# Choice

A. 15

B. 30

C. 45

D. 90

E. 15

F. Cannot be determined


Question 5a of 10

Based only on the given information, it is guaranteed that .

A. True

B. False

Question 6a of 10

Based only on the given information, it is guaranteed that

ABD CBD.

# Choice

A. True

B. False


Question 7a of 10


# Choice

A. True

B. False

Question 8a of 10


# Choice

A. True

B. False


Question 9a of 10

If a point is on the perpendicular bisector of a segment, then it is:

# Choice

A. equidistant from the midpoint and one endpoint of the segment.

B. on the segment.

C. equidistant from the endpoints of the segment.

D. the midpoint of the segment.

Question 10a of 10

If a point is on the bisector of an angle, then it is:

# Choice

A. equidistant from the two sides of the angle.

B. the vertex of the angle.

C. equidistant from the bisector and one side of the angle.

D. on one side of the angle.

apex algebra with trig and stats learning packet · 2020-04-10 · ll theorem again, the triangles...

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