Geometry ~ Chapter Four



Geometry ~ Chapter Four Name: _____________________

Congruent Triangles

4.1 Congruent Figures

|Congruent polygons: |**When you name congruent polygons, |

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|- If two angles of one triangle are congruent to two angles of another triangle, |

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|…but are the triangles congruent? |

1. [pic]

List four pair of congruent angles:

List four pair congruent sides:

|22. [pic] |3. It is given that [pic]. If [pic] what is [pic] Explain. |

|[pic] | |

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|4. Can you conclude that [pic] |5. [pic]. List the congruent corresponding parts. |

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|[pic] | |

|List corresponding vertices in order. | |

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|Can you conclude that the triangles are congruent | |

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To prove triangles are congruent, we need to show that the triangles contain 6 congruent parts.

4.2 Triangle Congruence by SSS and SAS

|Side-Side-Side Postulate |Side-Angle-Side Postulate |

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Decide whether you can use the SSS or SAS Postulate to prove the triangles are congruent. If so, write the congruence statement, and identify the postulate.

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|Given: M is the midpoint of [pic] |

|Prove: [pic] |

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|2. |

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|Given: H is the midpoint of [pic] |

|Prove: [pic] |

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|3. [pic] . What other information do you need in order to prove [pic]by SAS?|4. From the given information, can you prove [pic]? Explain. |

|[pic] | |

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|5. From the information given, can you prove [pic]? Explain. | |

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|Given: [pic] | |

|[pic] | |

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Prove the following:

4.3 Triangle Congruence by ASA and AAS

|Angle-Side-Angle Postulate |Angle-Angle-Side Postulate |

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|1. Using only the information in the diagram at the right, can|[pic] |

|you conclude that [pic] is congruent to either of the other |2. Suppose [pic] is congruent to [pic] and [pic] is not congruent to [pic] . Name the |

|two triangles? Explain. |triangles that are congruent by the ASA Postulate. |

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|3. |

|Given: [pic] [pic] |

|Prove: [pic] |

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|4. |

|Given: [pic] |

|Prove: [pic] |

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|5. |

|Given: [pic] |

|Prove: [pic] |

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4.4 Using Congruent Triangles: CPCTC

Once you have triangles that you have proven congruent, you can make statements about their other parts because

corresponding parts of congruent triangles are congruent : ___________.

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|Given: [pic] |

|Prove: [pic] |

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|2. |

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|Given: [pic] [pic] |

|Prove: [pic] |

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4-5 Discovery

Step 1: On a separate piece of paper, construct an Isosceles Triangle with a compass

Step 2: Name your triangle ABC with A and B being on opposite verticies of the congruent sides

Step 3: Fold the paper so the two congruent sides fit precisely on top of each other. Crease the paper. Label the intersection of the fold line and AB as point D

Questions:

1. What do you notice about angles A and B? Make a conjecture about the angles opposite the congruent sides in an Isosceles Triangle.

2. Study the fold like CD and the base AB.

a) As best as you can tell, what type of angle is CDA? Angle CDB?

b) How do AD and BD seem to be related?

c) Use your answers to parts (a) and (b) to complete the conjecture:

The fold line CD is the ____________of the base AB of isosceles triangle ABC

Quick check 1

4.5 Isosceles and Equilateral Triangles

|Draw an isosceles triangle. |

|Label the vertex angle, the base, |

|the legs, and the base angles. |

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Draw an isosceles triangles. Label the vertex angle, the base, the legs, and the base angles.

Three things we know about isosceles triangles:

|Theorem 4-3 | |

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|Theorem 4-4 | |

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|Theorem 4-5 | |

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|Draw an equilateral triangle: |

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|If we know that all three sides are the same length in a triangle, then the angles must all be the same measure…and if we know that the angles are all the |

|same measure in a triangle, then the sides must be the same length. |

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|_________________________ - the angles all have the same measure |

|(and in a triangle, what must that measure be? ________ ) |

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|_________________________ - all of the sides have the same length |

|1. Explain why [pic] is isosceles. |2. Find x and y. |

|[pic] |[pic] |

|3. Find the value of t. |4. Find the value of x, y, and z. |5. Find the value of x and y. |

| |[pic] |[pic] |

|[pic] | | |

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|Find x | | |

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Draw an equilateral triangle:

If we know that all three sides are the same length in a triangle, then the angles must all be the same measure…and if we know that the angles are all the same measure in a triangle, then the sides must be the same length.

_________________________ - the angles have the same measure (and in a triangle, what must that measure be? ____)

_________________________ - the sides have the same length

Example1, 2

4.6 Congruence in Right Triangles

|Hypotenuse- | |

|Leg (HL) | |

|Theorem | |

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|1. |

|Given: [pic]are right angles, [pic] |

|Prove: [pic] |

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|2. One student wrote, “[pic]by the HL Theorem” for the diagram. Is the |3. Which two triangles are congruent by the HL Theorem? Write a correct |

|student correct? Explain. |congruence statement. |

| |[pic] |

|[pic] | |

|4. You know that two legs of one right triangle are congruent to two legs |5. What additional piece of information do you need in order to prove the |

|of another right triangle. Explain how to prove the triangles are |triangles congruent using HL? |

|congruent. |[pic] |

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Example1, 3 Quick Check 1, 3

More Proofs:

4.7 Using Corresponding Parts of Congruent Triangles

It definitely helps to re-draw individual triangles when you are asked to prove triangle congruence! Don’t write in your book!

Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL.

|1. Given: [pic] |2. Given: [pic] |

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| |[pic] |

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|3. Separate and redraw the indicated triangles. Identify any common angles |4. Separate and redraw the indicated triangles. Identify any common angles |

|or sides. |or sides. |

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|[pic] |[pic] |

|[pic] |[pic] |

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Ready for the big one?

5. Given: [pic]

Prove: [pic]

Example 1, 4 Quick Check 1

More Proofs:

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