Geometry S1 Unit 4

Hug HS Math Department: Lesson based on units

Proctor Hug HS Geometry S1 Unit #4

List of Teachers

Evan Brandt

Arthur Pacheco

Carol Mischel

Unit 5 Notes: Congruence and Triangles

This packet is your notes for all of Unit 5. It is expected you will take good notes and work the examples in class. It is expected that you bring your packet to class every day and do not lose it! Should you be absent, it is expected that you get the notes and examples you missed. This packet will be collected and graded on the day of the Unit 5 Test.

5.1: Angles of a Triangle

Warm up: Find all missing angles

*Cut out a triangle. Fold as shown below.

What have you discovered?

The sum of the angles in a triangle is ________________________________________.

Triangle Sum Theorem

The sum of the measures of the interior angles of a triangle is __________°.

mA + mB + mC =______°. A

B

C A B C

B C

Example 1: Find mY in triangle RST.

Example 2: Find the values of x and y in the figure.

Example 4: Find the value of each angle in the triangle.

Objective #3: Can you explain in words how you could use the Triangle Sum Theorem to solve for a variable?

Right Triangle Angle Sum Theorem: The acute angles of a right triangle are ____________________________.

_________ + _________ = 90°

x°

Example 6: Find m A and m B in the right triangle ABC.

Triangle Inequality Theorem

Get 2 Spaghetti noodles, break one of the noodles, and make a triangle and draw your results.

http://www.mathopenref.com/triangleinequality.html

TI Theorem: side + side > 3

side

Example Non – Example

Example Non – Example

3x°

2x° A

5.2: Classifying Triangles Warm up:

State whether the angles are acute, right, obtuse, or straight.

1) 90° 2) 3) 87°

A triangle is a figure formed when _____ _____________________ are connected by segments. A triangle

can be classified by its __________ and by its ___________.

Parts of a triangle:

Vertex (plural: vertices) - Each of three _________ joining the _________ of a triangle.

Adjacent sides – Two sides sharing a common _________________.

Classified by Angles

Acute – all acute angles

Classified by Sides

Scalene – no congruent sides

Isosceles – at least 2 sides congruent Equilateral – all sides congruent

Example: Classifying Triangles

What are all the possible classifications for each of these triangles? (remember angles and sides)

a) b) c)

Objective : Can you classify a triangle by its sides and angles?

Classify the triangles:

a) b) ₃ c)

d) Create a triangle and mark measurements for the sides and angles. Trade papers with a friend and have him/her classify your triangle.

Example: Find the lengths of AB and BC of isosceles triangle ABC if A is the vertex angle.

Example: In the following diagram, , and . What is the measure of ?

A. 30 B. 120 C. 150 D. 165

Example: Find x and y.

A

23 5x-7

3x-5 C

Objective: Can you find the measures of AC and BC of the following isosceles triangles if angle A is the vertex angle?

5.3: Congruent Triangles Warm up:

1)Name the angle in 4 different ways: 2) Solve for x, y, and z

When two triangles are congruent, there are 3 pairs of corresponding angles that are congruent and 3 pairs of corresponding sides that are congruent.

Corresponding Angles

Corresponding Sides

Congruence statement – tells us the corresponding parts of the two triangles (which parts are congruent).

Example 1: The corresponding parts of two congruent triangles are marked on the figure.

List the congruent angles and sides.

Write a congruence statement.

Example 2:

Find the value of x. (Not drawn to scale)

Objective : Can you state which side or angle is congruent to the listed part if RST ≅ XYZ?

a)  R ≅ _______ b) RS≅ ________ c) ΔSRT ≅ _____

 Y ≅ _______ ST≅ ________

 T ≅ _______ TR≅ ________

E

D F

B

87 _{4x + 11}

22

Objective : Can you explain what it means when one triangle is congruent to another?

Objective : Can you complete the congruence statement?

ABD ≅ ________

Objective : Can you write a congruence statement for the pair of triangles represented by the congruent parts? (Draw a picture!)

a)

Warm up:

Cross out the values that do not make sense for x.

x = 0 x = 1 x = 2 x = 3 x = 4

Why do those values not make sense?

Side-Side-Side Congruence Postulate (SSS) - If three sides of one triangle are congruent to _______ sides of a second triangle, then the two triangles are congruent.

____≅ _____ ____≅ _____

____≅ _____ so by SSS ABC _____∆ ABC≅∆≝¿.

Side-Angle-Side Congruence Postulate (SAS) – If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

If there’s only one included angle, it must be between the 2 given sides. There’s no congruence for SSA.

______ ≅ ______

______≅ ______

Example:

I. Decide if there is enough information to prove that the triangles are congruent. II. If there is enough information, state the congruence postulate you would use.

a) b)

c)

Example: Write a congruence statement for the pair of triangles represented by the following statements, if possible. (Draw a picture!)

a) b)

5.5: ASA and AAS Congruence

Warm up:

1. Refer To the figure to complete the congruence statement, ∆ ABC≅¿ by _______

Angle-Side-Angle Congruence Postulate (ASA) – If two angles and the _____________ side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

_____≅ _____ ______ ≅______

_____≅ ______ so then by ASA

ABC  ______∆ ABC≅∆≝¿

CPCTC –Corresponding Parts of Congruent Triangles are Congruent.

- If two triangles are congruent, then so are their corresponding parts. Given HUK VTJ

List the congruent sides and angles.

K U

H

J T

Example 6: Given CAT  DOG, which of the following are true?

 C  D

 T  G

A D  

C

A T

D

G

5.6 HL Theorem: Warm up:

1. Which theorem can be used to conclude that ∆ CAB≅∆ CED?

Hypotenuse Leg (HL) – If the Hypotenuse and a leg are congruent then the triangles are congruent.

_____ ≅ _____ ______ ≅______

ABC  ______ by _______∆ ABC≅∆≝¿

Your Practice: Write a congruence statement if possible:

A) B)

Geometry Name ___________________________________________ Period ______

Unit 5: Worksheet 1

Find the missing angle measurement for each triangle:

State if the three numbers can be the measures of the sides of a triangle.

9) 5, 2, 4 10) 8, 2, 8

11) 9, 6, 5 12) 5, 8, 4

13) 4, 7, 8 14) 11, 12, 9

Review:

15. Determine the slope of this line: y = 2x -4

16. Write an equation of a line parallel to the line in #15 with a y intercept of 3.

17. Write an equation of a line perpendicular to the line in #15 with a y intercept of -2

Geometry Name _____________________________ Per ____

19.

22. Determine if the lines are parallel, perpendicular or neither.

A. y = 2x – 4 B. y = 2x – 4 C. y = 2x – 4 D. y = 2x – 4

y = 4x + 2 y = 2x + 3 y = - ½x + 4 y = 2x - 4

Geometry Name ____________________________________ Per ____ Unit 5 Worksheet 3

Complete Each Congruence Statement

1.

3. 5.

6.

7. Solve for x

10. Which equation of the line passes through (6,−8) and is perpendicular to the graph of the line y=2

3x−4?

11. Write the equation of the line passes through (4,7) and is parallel to the graph of the line that passes through the points (1,3) and (−2,9) ?

Solve for the variable.

20. If ∆ CED≅∆ QRP, which of the following is true? A

.

∠C≅∠Q ,∠E≅∠R ,∠D≅∠P

B .

∠C≅∠Q ,∠E≅∠P ,∠D≅∠R

C .

∠C≅∠P ,∠E≅∠R ,∠D≅∠Q

D .

∠C≅∠R ,∠E≅∠Q ,∠D≅∠P

21. In the figure ∠GAE≅∠LOD and AE≅DO. What information is needed to prove that ∆ AGE≅∆ OLD by SAS?

A .

¿≅LD B

.

AG≅OL

C .

∠AGE≅∠OLD

D .

∠AEG≅∠ODL

1.

2.

3.

4. 5.

6. 7.

Geometry Name ____________________________________ Per ____

Unit 5 Worksheet 6

State whether the given side is a leg or a hypotenuse.

Determine whether you can use the HL Congruence Theorem to show that the triangles are congruent (yes or no). Then Explain your reasoning.

7. 8. 9.

__________________________ _____________________________ ____________________________ __________________________ _____________________________ ____________________________ __________________________ _____________________________ ____________________________ __________________________ _____________________________ ____________________________

11. In the figure ∠GAE≅∠LOD and AE≅DO. What information is needed to prove that ∆ AGE≅∆ OLD by AAS?

A .

¿≅LD B

.

AG≅OL

C .

∠AGE≅∠OLD

D .

∠AEG≅∠ODL

12. Which conclusion can be drawn from the given facts in the diagram? A

.

TQ bisects ∠PTS B

.

∠TQS≅∠RQS

C .

PT≅RS

D .

TS=PQ

A .

∆ HIJ≅∆ LKJ by ASA B

.

∆ HIJ≅∆ KLJ by SSS C

.

∆ HIJ≅∆ KLJ by SAS D

.

∆ HIJ≅∆ LKJ by SAS

Geometry Name_______________________________ Per_____ Unit 5 Practice Test

1.)Which of the following terms can be used to describe a triangle that has two equal sides?

A. Equilateral C. Scalene B. Acute D. Isosceles

For # 2,3 classify each triangle by its angles and sides.

2.) 3.)

4.) find the angle with the ?

5.) Solve for the variable

8.)Which statement correctly describes the congruence of the triangles shown in the diagram?

A. Δ ABC≃ΔCDA

B. Δ ABC≃Δ ADC

C. Δ ABC≃Δ DCA

D. Δ ABC≃Δ ACD

6. )

If ∆ CED≅∆ QRP, list the congruent angles and sides.

A D B C Answers 7. 8. 9.

7.) Refer To the figure to complete the congruence statement,

11. )

Determine which postulate or theorem can be used to prove the pair of triangles congruent.

12.

)

Which of the following sets of triangles cannot be proved congruent? There may be more than one.

A. C.

B. D.

13. )

In the figure ∠A≅∠O and

AE≅DO. What information is needed to prove that

∆ AGE≅∆ OLD by ASA?

34 Answers 10. 11. 12. 13 10. )

Which theorem can be used to conclude that

Answers

14.

15.

x = 16. 14.) In the figure ∠H≅∠L and IH≅KL. Write a

congruence statement.

15. In the figure, write a congruence statement, and solve for x.

16. If a triangle has two sides with lengths of 8cm and 14cm. Which length below could not

represent the length of the third side?

A. 7cm C. 15cm

B. 13cm D. 22cm

For #17, 18 use the following:

Given: Q is the midpoint of MN; ∠MQP≅∠NQP

Prove: ∆ MQP≅∆ NQP

17 .

Choose one of the following to complete the proof.

A. MN≅QP

B. MQ≅NQ

C. MP≅NP

D. QP≅QP

18

. Choose one of the following to complete the proof. A. Reflexive property of equality

B. SSA Congruence

C. SAS Congruence

D. AAS Congruence

19

. In the figure,

∆ MON≅∆ NPM. Solve for x and y

36

Statements Reasons

Q is the midpoint of MN; ∠MQP≅∠NQP Given

17. Definition of Midpoint

∠MQP≅∠NQP Given

QP≅QP Reflexive property of congruence