Unit 3: Triangles and Polygons - MR. DAVISridavis.weebly.com/uploads/8/7/2/6/87260484/unit_3_notes.pdf · Example 3: How much information do we need before knowing that two triangles

Unit 3: Triangles and Polygons

Background for Standard G.CO.9: Prove theorems about triangles.

Objective: By the end of class, I should…

Example 1: Trapezoid on the coordinate plane below has the following vertices: 3,8 , 3,4 ,

11,4 , and 11,10 . Shade the interior of this shape with your pencil.

was translated 14 units to the left and 12 units down

to form trapezoid ′ ′ ′ ′. List the coordinates of the new vertices.

′: ′: ′: ′:

Is ≅ ′ ′ ′ ′? Explain your reasoning.

was rotated 90° counter‐clockwise about the origin

to form trapezoid . List the coordinates of the new

vertices.

: :

: :

Is ≅ Explain your reasoning.

was reflected across the x‐axis to form trapezoid . List the coordinates of the new vertices.

: :

: :

Is ≅ ? Explain your reasoning.

**In order for two shapes to be congruent (exactly the same), corresponding angles must be congruent

and corresponding sides must be congruent.

Example 2: Consider the congruence statement:∆ ≅ ∆

A. Identify the congruent angles. B. Identify the congruent sides.

A

B C

D

A’

B’ C’

D’

I

J K

L

E F

G H

Example 3: How much information do we need before knowing that two triangles are congruent?

A. Given three fixed side lengths, how many different triangles can you create?

What pattern is this? ______________ Is it a triangle congruency theorem? __________

B. Given two fixed side lengths and a fixed angle (where the angle is in between the sides, how many

different triangles can you create?


C. Given one fixed angle and one fixed side length, how many different triangles can you create?


D. Given two fixed angles and one fixed side length (where the side is between the two angles), how

many different triangles can you create?


E. Given two fixed side lengths, how many different triangles can you create?


F. Given two fixed side lengths and one fixed angle (where the angle is not in between the sides), how

many different triangles can you create?


G. Given three fixed angles, how many different triangles can you create?


H. Given two fixed angles and one fixed side (where the side is not in between the two angles), how many

different triangles can you create?


Use the previous page to summarize your findings about triangle congruence theorems.

Triangle Congruence Theorems NOT Triangle Congruence Theorems

How many pieces of information are necessary to have congruent triangles?

What two patterns with three pieces of given information are NOT theorems? If not congruency,

what can they guarantee?

Example 4: Suppose ≅ and ≅ in the diagram shown. Are there congruent triangles in the

diagram? If so, write a triangle congruence statement and name the theorem used.

Example 5: Suppose is the midpoint of and is the midpoint of in the diagram shown. Are there

congruent triangles in the diagram? If so, write a triangle congruence statement and name the theorem

used.

Example 6: Suppose ≅ , and bisects ∠ in the diagram shown. Are there congruent triangles in

the diagram? If so, write a triangle congruence statement and name the theorem used.

Use the diagrams below and the provided information to prove the two triangles are congruent by SSS, SAS, ASA, and AAS. Example 7: Example 8: Example 9: Given and ∠ ≅ ∠ Example 10:

Statement Reason

Statement Reason

Statement Reason

Statement Reason



List the 4 triangle congruence theorems you explored previously.

These congruence theorems apply to all triangles. There are also theorems that only apply to right

triangles. To prove that two right triangles are congruent, only two pieces of information (side lengths or

angles) are necessary because you are always given one angle—the right angle.

Which general triangle theorem (listed in the boxes at the top of this page) correlates with each right

triangle theorem?

Leg‐Leg (LL) Congruence Theorem—

Hypotenuse‐Angle (HA) Congruence Theorem—

Leg‐Angle (LA) Congruence Theorem—

Hypotenuse‐Leg (HL) Congruence Theorem—

*Because most of the right triangle congruency theorems are repeats of the general triangle congruency

theorems, the only one we really care about is _______________. Add this special theorem to your list

above with an asterisk (*).

Explain why only two pairs of corresponding parts are needed to prove that two right triangles are

congruent.

Use the diagrams below and the provided information to prove the two right triangles are congruent. Example 1:

Statement Reason

Example 2:

Example 3:

Example 4: Given: , ≅ Prove: :∆ ≅ ∆

Review : What kind of triangle is GBD? What are the congruent base angles of triangle GBD?

What does it mean for two triangles to be congruent?

Corresponding _____________ must be congruent and corresponding _____________ must be congruent.

Therefore, if two triangles are congruent, then ________________________________________________.

This is what’s known as C.P.C.T.C.

Statement Reason

Statement Reason

Statement Reason

Example 6: In the diagram below, is ≅ ? Explain your reasoning.

Example 7: In the diagram below, is ∠ ≅ ∠ ? Explain your reasoning.

Example 8: In the diagram below, ≅ , ≅ , and ∠ ∠ are right angles. Is ∠ ≅∠ ? Explain your reasoning.

Statement Reason

Statement Reason

Statement Reason

To use CPCTC to explain your reasoning, follow these steps:

Step 1: Identify two triangles in which segments or angles are corresponding parts.

Step 2: Use a triangle congruency theorem to prove the triangles congruent.

Step 3: State that the two parts are congruent using CPCTC as the reason.



The Isosceles Triangle Base Theorem states:

The altitude to the base of an isosceles triangle bisects the base.

Draw an altitude from the vertex angle to the base. Remember altitudes are perpendicular to bases.

The altitude to the base also ________________ the vertex angle. The altitude is ____________________ to the

base.

In an isosceles triangle, the altitudes to the congruent sides are ______________.

In an isosceles triangle, the angle bisectors to the congruent sides are

_____________.

Example 1: Given: , ≅ , 12 . , 20 . Find the width of the doghouse.

Logic Statements

Conditional: If p, then q.

Converse: If q, then p.

Inverse: If not p, then not q.

Contrapositive: If not q, then not p.

Truth Value: True means always true!

Example 2: Given the following conditional statement, complete the remaining logic statements and give the truth

value for each.

Conditional: If a triangle is equilateral, then it is isosceles. T/F? ___

Converse: T/F? ___

Inverse: T/F? ___

Contrapositive: T/F? ___

Example 3: Create a conditional statement that is not related to Geometry.

Example: If my alarm goes off, then I get out of bed.

Write out the converse, inverse and contrapositive. Then give the truth value for all four statements.

When you have completed this, compare your example with a neighbor and be prepared to share your example with

the class. Be creative!

Conditional: T/F? ___

Converse: T/F? ___

Inverse: T/F? ___

Contrapositive: T/F? ___

A Biconditional Statement (if and only if) is true only when the conditional and the converse are both true.

Example: A trapezoid is isosceles if and only if its diagonals are congruent.

Is the conditional true? If the converse true?

Background for Standard G.CO.11: Properties of Squares, Rectangles, Parallelograms, Rhombi, Kites and

Trapezoids.


Polygon: a two‐dimensional, plane shape that is made of straight lines and is “closed” (all the lines

connect). The root of the word “polygon” is Greek: Poly‐ means “many” and –gon means “angle.” So

many angle—there you go!

Quadrilateral: a four‐sided polygon

We will discuss 7 different quadrilaterals: Squares, Rectangles, Parallelograms, Rhombi, Kites, Trapezoid,

and Isosceles Trapezoid. (Refer to your shapes for details about each quadrilateral).

Some of the shapes on the previous page are special cases of the other shapes. This flowchart details

which categories are subset of which other categories.

Example 1: Name at least one quadrilateral that fit each description.

A. Both pairs of opposite sides are congruent

B. All angles are congruent

C. Each diagonal bisects opposite angles

D. Exactly one pair of opposite sides are parallel

E. Consecutive angles are supplementary

F. Exactly one pair of opposite angles are congruent

G. Exactly one pair of opposite sides are parallel and each pair of base angles are congruent

H. All sides are congruent

Example 2: Create a flowchart to classify any quadrilateral based on the filled‐in questions. This chart will

help you categorize any quadrilateral.

How many pairs of

parallel sides are there?

Are there four

right angles?

Are the four

sides congruent? Are the four

sides congruent?

0

1

2

Yes No

Yes

No Yes

No

Background for Standard: Use geometric shapes, their measures, and their properties to describe objects.


Interior Angles An interior angle faces the inside of the polygon and is formed by consecutive sides of the polygon. Let’s

start with the polygon with the fewest number of sides (a triangle) and work up to a polygon with 10 sides

(called a decagon) and look at the sum of all of the interior angles.

Triangle (3 sides) Pentagon (5 sides)

Sum of Interior Angles: Number of Triangles:

Sum of Interior Angles:

Quadrilateral (4 sides) Hexagon (6 sides)

Number of Triangles: Number of Triangles:

Sum of Interior Angles: Sum of Interior Angles:

Heptagon (7 sides) Nonagon (9 sides)



Octagon (8 sides) Decagon (10 sides)



What is the formula for the sum of the interior angles of an n‐sided polygon? Explain your reasoning.

Example 1: What is the sum of all of all the interior angle measures of a 100‐sided polygon?

Example 2: If the sum of all the interior angle measures of a polygon is 9540°, how many sides does the

polygon have?

Exterior Angles The picture on the right shows what exterior angles

look like on a triangle. Each exterior angles forms a

linear pair with an interior angle. Let’s see if we can

come up with a formula for the sum of all of the

exterior angle measures of a polygon, starting with a

triangle.

Triangles: Calculate the sum of the exterior angle measures of a triangle with the following steps.

1. In the provided triangle estimate the degree measure of each angle. You can estimate these numbers,

but make sure they add up to 180° like all triangles do.

2. Use the angle measures you estimated to find the measure of each exterior angle.

Remember that each interior angle is supplementary to its exterior angle.

3. Add up the three exterior angles.

Quadrilaterals: Repeat the steps above to calculate the sum of the exterior angle measures

1. In the provided quadrilateral estimate degree measure of each angle. You can estimate these numbers,

but make sure they add up to 360° like all quadrilaterals do.

2. Use the angle measures you estimated to find the measure of each

exterior angle. Remember that each interior angle is supplementary to its

exterior angle.

3. Add up the four exterior angles.

Are you seeing a pattern? What is the sum of all of the exterior angle measures of a polygon?

Example 3: Use what you know about the sum of the exterior angles of a polygon to solve for the

variable(s) in each picture.

A. B.

Regular Polygons In a regular polygon, all of the sides are congruent. This means that all of the angles are also congruent.

If all of the angles of a regular polygon are congruent, then we can find the measure of one interior or

exterior angle by dividing the formula by the number of angles .

Measure of each interior angle of a regular polygon:

180 2

Measure of each exterior angle of a regular polygon:

360

Example 4: Use the formula to calculate each interior angle measure of a regular 100‐sided polygon.

Example 5: If each interior angle measure of a regular polygon is equal to150°, determine the number of

sides. Explain how you calculated your answer.

Example 6: Calculate the measure of each exterior angle of an equilateral triangle (a regular triangle).

Explain your reasoning.

Example 7: Calculate the measure of each exterior angle of a 25‐sided polygon. Explain your reasoning.

Example 8: If the measure of each exterior angle of a regular polygon is 18°, how many sides does the

polygon have? Explain how you calculated your answer.

Review

Example 1: With your partner, complete the table by placing a checkmark in the appropriate row and column to associate each figure with its properties.

Example 2: As a class, create a Venn diagram that describes the relationships between all of the quadrilaterals listed: trapezoid, rhombi, parallelograms, quadrilaterals, kites, rectangles, squares.

Example 3: True or False? ___ A square is also a rectangle. ___ A rectangle is also a square. ___ The base angles of a trapezoid are congruent. ___ A parallelogram is also a trapezoid. ___ A square is a rectangle with all sides congruent. ___ The diagonals of a trapezoid are congruent. ___ A kite is also a parallelogram. ___ The diagonals of a rhombus bisect each other.

Characteristic Quad

rilateral

Trapezo

id

Kite

Parallelogram

Rhombus

Rectan

gle

Square

No parallel sides

Exactly one pair of parallel sides

Two pairs of parallel sides

One pair of sides are both congruent and parallel

Two pairs of opposite sides are congruent

Exactly one pair of opposite angles are congruent

Two pairs of opposite angles are congruent

Consecutive angles are supplementary

Diagonals bisect each other

All sides are congruent

Diagonals are perpendicular to each other

Diagonals bisect the vertex angles

All angles are congruent

Diagonals are congruent

Documents

Unit 3: Triangles and Polygons - MR. DAVISridavis.weebly.com/uploads/8/7/2/6/87260484/unit_3_notes.pdf · Example 3: How much information do we need before knowing that two triangles