4.1 Apply Triangle Sum Properties Geometry

Name: __________________________ Date________Period: 1 2 3 4 5

Interior Angles

The angles INSIDE any polygon formed by the intersection of any two sides.

Exterior Angles

The angles OUTSIDE any polygon formed by extending the sides of the polygon.

Each exterior angle will be a linear pair with its interior angle. Triangle Sum The sum of the measures of the three interior Theorem angles of any triangle is 180°. Corollary to Triangle Sum Thm. Exterior Angle Theorem

The acute angles of a right triangle are always complementary. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles (or nonadjacent angles) of the triangle.

Classify Triangles by Sides: Scalene No sides congruent. Triangle Isosceles Triangle

Two sides congruent.

Equilateral Triangle

All sides congruent.

Classify Triangles by Interior Angles: Acute All angles acute. Triangle Right Triangle

One angle = 90°.

Obtuse Triangle

One angle obtuse.

Equiangular Triangle

All angles congruent.

Example 1) What type of triangle is the scoring triangle in shuffleboard? How do you know?

Checkpoint 1) Draw: a) An isosceles right triangle. b) An obtuse scalene triangle.

Example 2) Classify △ 𝑅𝑆𝑇 by its side lengths. Then determine if it is a right triangle. Step 1) Use the ______________________ to find side lengths.

Step 2) Use the _____________________ to find any right angles.

Example 3) Find the measure of ∠𝐷𝐶𝐵.

Example 4) The side of the wheelchair ramp shown forms a right angle. The measure of one acute angle in the right triangle is eight times the measure of the other. Find the measure of each angle.

Checkpoint 2): △ 𝐽𝐾𝐿 has vertices J (-2, -1), K (1, 3) and L (5, 0). a) Classify △ 𝐽𝐾𝐿 by its sides.

b) Then determine if it is a right triangle.

Checkpoint 3): Find the measure of ∠1.

4.2 Apply Congruence & Triangles Geometry

Name: _________________________ Date________Period: 1 2 3 4 5 6

Congruent Figures

If two figures are congruent, then all of their corresponding parts will be congruent.

Congruence Statement

The statement naming two figures as congruent. Corresponding parts must be named in the same order. (this is a big deal!)

Third Angles If two angles of one triangle are congruent to Theorem two angles of another triangle, then the third (“No angles will also be congruent. Choice” Thm.)

Properties of Congruent Triangles: Reflexive

Symmetric

Transitive

Example 1) Write a congruence statement for the triangles. Identify all corresponding congruent parts.

Example 2) In the diagram 𝑄𝑅𝑆𝑇 ≅ 𝑊𝑋𝑌𝑍. a) Find the value of x.

b) Find the value of y.

Use the same diagram for checkpoints 1 & 2: Checkpoint 1) 𝐹𝐺𝐻𝐽 ≅ 𝑆𝑇𝑈𝑉. Name all pairs of corresponding congruent parts.

Checkpoint 2) 𝐹𝐺𝐻𝐽 ≅ 𝑆𝑇𝑈𝑉. Find the value of x and the measure of ∠𝐺.

Example 4) Find 𝑚∠𝑉.

̅̅̅ Example 5) Given: ̅̅̅̅ 𝐹𝐻 ≅ ̅𝐽𝐻 ∠𝐹𝐻𝐺 ≅ ∠𝐽𝐻𝐺 ̅̅̅̅ ̅̅̅ 𝐹𝐺 ≅ 𝐽𝐺 ∠𝐹𝐺𝐻 ≅ ∠𝐽𝐺𝐻 Prove: △ 𝐹𝐺𝐻 ≅△ 𝐽𝐺𝐻

̅̅̅̅ & 𝐵𝐷 ̅̅̅̅ . Checkpoint 3): E is the midpoint of 𝐴𝐶 Show that △ 𝐴𝐵𝐸 ≅△ 𝐶𝐷𝐸. (proof)

4.3 Prove Triangles Congruent by SSS Geometry

Side-SideSide (SSS) Congruence Postulate

Name: __________________________ Date________Period: 1 2 3 4 5 6

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

̅̅̅ ≅ 𝐽𝐻 ̅̅̅̅ Example 1) Given: 𝐽𝐹 ̅̅̅̅ G is the midpoint of 𝐹𝐻 Prove: △ 𝐹𝐽𝐺 ≅△ 𝐻𝐽𝐺

Example 2) Determine whether △ 𝑃𝑄𝑅 is congruent to any of the other triangles. ̅̅̅̅ ______ Step 1) Find the length of ̅̅̅̅ 𝑃𝑄 ______ Find the length of 𝑄𝑅 Find the length of ̅̅̅̅ 𝑃𝑅 _____________________

̅̅̅̅ ______ Find the length of 𝑅𝑇 ̅̅̅̅ ______ Step 2) Find the length of 𝑅𝑆 ̅̅̅̅ _____________________ Find the length of 𝑆𝑇

̅̅̅̅̅ _____________________ Step 3) Find the length of 𝑉𝑊

Find the length of ̅̅̅̅̅ 𝑅𝑊 _____________________

Find the length of ̅̅̅̅ 𝑉𝑅_____________________

Example 3) Explain why the table with the diagonal legs is stable, but the table with vertical legs could collapse.

Checkpoint 1) Decide whether the congruence statement is true. Explain. a) △ 𝐽𝐾𝐿 ≅△ 𝑀𝐾𝐿

b) △ 𝑅𝑆𝑇 ≅△ 𝑇𝑉𝑊

Checkpoint 2): △ 𝐷𝐹𝐺 has vertices D (-2, 4), F (4, 4), G (-2, 2). △ 𝐿𝑀𝑁 has vertices L (-3, -3), M (-3, -3), N (-1, -3). Graph the triangles and show that they are congruent.

Checkpoint 3) Decide whether the figure is stable. Explain your reasoning. a)

b)

4.4 Prove Triangles Congruent by SAS & HL Geometry

Name: _____________________ Date_______Period: 1 2 3 4 5 6

Side-AngleSide (SAS) Congruence Postulate

If two sides and the INCLUDED angle of one triangle are congruent to two sides and the INCLUDED angles of another triangle, then the two triangles are congruent.

HypotenuseLeg Congruence Theorem

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

̅̅̅ ≅ ̅̅̅̅ Example 1) Given: ̅𝐽𝑁 𝐿𝑁 ̅̅̅̅̅ ̅̅̅̅̅ 𝐾𝑁 ≅ 𝑀𝑁 Prove: △ 𝐽𝐾𝑁 ≅△ 𝐿𝑀𝑁

Example 2) ABCD is a rectangle. What can you conclude about △ 𝐴𝐵𝐶 𝑎𝑛𝑑 △ 𝐶𝐷𝐴? Explain.

Example 3) Given: ̅̅̅̅ 𝐴𝐶 ≅ ̅̅̅̅ 𝐸𝐶 ̅̅̅̅ ̅̅̅̅ 𝐴𝐵 ⊥ 𝐵𝐷 ̅̅̅̅ ̅̅̅̅ 𝐸𝐷 ⊥ 𝐵𝐷 ̅̅̅̅ 𝐴𝐶 is a bisector of ̅̅̅̅ 𝐵𝐷 Prove: △ 𝐴𝐵𝐶 ≅△ 𝐸𝐷𝐶

Example 4) The gate to a ranch is a rectangle (ABDE). You know that △ 𝐴𝐹𝐶 ≅△ 𝐸𝐹𝐶. How could you conclude that △ 𝐴𝐵𝐶 ≅△ 𝐸𝐷𝐶?

Checkpoint 1) Explain why the diagonal of a rectangle always forms two congruent triangles.

Checkpoint 2) In example 4, suppose you know that ABCF and EDCF are squares. How could you prove △ 𝐴𝐵𝐶 ≅△ 𝐸𝐷𝐶?

4.5 Prove Triangles Congruent by ASA & AAS Name: __________________ Geometry

Date______Period: 1 2 3 4 5 6

Angle-SideAngle (ASA) Congruence Postulate

If two angles and the INCLUDED side of one triangle are congruent to two angles and the INCLUDED side of another triangle, then the two triangles are congruent.

Angle-AngleSide (AAS) Congruence Theorem

If two angles and a NON-included side of one triangle are congruent to two angles and the corresponding NON-included side of another triangle, then the two triangles are congruent.

Side Required!

What do all of the postulates & theorems about triangle congruency have in common?

Example 1) Can you prove the triangles congruent? Why or why not?

Checkpoint 1) Can you prove △ 𝑆𝑇𝑊 ≅△ 𝑉𝑊𝑇 given the diagrams as marked? If so, state the postulate or theorem that you would use. a)

b)

Example 2) Write a flow proof:

Example 3) You & a friend are trying to find a flag hidden in the woods. You & your friend are 75 ft. apart. When you are facing each other, the angle formed by you and the flag is 72° and the angle formed by your friend & the flag is 53°. Is there enough information to find the flag?

Checkpoint 2) Two actors stand 20 ft. apart on the stage in a theater. Two spotlights mounted on the ceiling shine on the actors and the spotlights are angled at 40° from each other. Can one actor move without out requiring the spotlight to move and without changing distance from the other actor?

4.6 Use Congruent Triangles (CPCTC) Geometry

Name: ___________________________ Date________Period: 1 2 3 4 5 6

Corresponding If two triangles can be proved congruent, then Parts of all of their corresponding parts must also be Congruent congruent. Triangles are Congruent (CPCTC) Example 1) Using the given information explain (in detail) how to prove △ 𝐴𝐵𝐶 ≅△ 𝐷𝐸𝐶. ̅̅̅̅ ≅ 𝐴𝐶 ̅̅̅̅ . Then use the fact that the triangles are congruent to prove that 𝐷𝐶 ̅̅̅̅ ≅ ̅̅̅̅ Given: 𝐴𝐵 𝐷𝐸 & ∠1 ≅ ∠2

Example 2) Use congruent triangles to find the distance between two docked boats (A & B). Step 1) Step 2) Step 3) Step 4)

Does it matter how far away from pt. B that you mark pt. D?

Checkpoint 1) Explain how to prove ̅̅̅̅ 𝑃𝑅 ≅ ̅̅̅̅ 𝑄𝑆.

Example 2) First use the information given to prove △ 𝐵𝐸𝐷 ≅△ 𝐶𝐸𝐷. Then use those congruent triangles to prove △ 𝐴𝐵𝐷 ≅△ 𝐴𝐶𝐷.

Checkpoint 2) ̅̅̅̅ ≅ 𝐾𝐽 ̅̅̅̅ ≅ 𝐿𝐾 ̅̅̅, 𝐹𝐺 ̅̅̅̅ Given: 𝐺𝐻 ∠𝐹𝐺𝐽 & ∠𝐿𝐻𝐾 are Rt. Angles Prove: △ 𝐹𝐽𝐾 ≅△ 𝐿𝐻𝐺.

Example 3) Write a proof to verify that the construction for copying an obtuse angle is valid. (Prove ∠𝐷 ≅ ∠𝐴)

4.7 Use Isosceles & Equilateral Triangles Geometry

Base Angles Theorem

Converse of Base Angles Theorem Corollary to Base Angles Thm. Corollary to Converse of Base Angles Thm.

Name: _______________________ Date________Period: 1 2 3 4 5 6

If two sides of a triangle are congruent, then the angles opposite them (the base angles) are congruent. (If sides ≅, then angles ≅.) If two angles (base angles) of a triangle are congruent, then the sides opposite them are congruent. (If angles ≅, then sides ≅.) If a triangle is equilateral, then it is also equiangular. If a triangle is equiangular, then it is also equilateral.

Example 1) Identify two congruent angles.

Example 2) Find the measure of ∠𝑅, ∠𝑆, & ∠𝑇.

Example 3) Find the values of x & y.

Example 4) The diagram shows a piece of quilting. a) Explain why △ 𝐴𝐷𝐶 is equilateral.

b) Show that △ 𝐶𝐵𝐴 ≅ △ 𝐴𝐷𝐶.

Checkpoint 1) In example 4 above, show that 𝑚∠𝐵𝐴𝐷 = 120°.

Checkpoint 2) If ̅̅̅̅ 𝐹𝐻 ≅ ̅̅̅ 𝐹𝐽, then which angles are congruent?

Checkpoint 3) If △ 𝐹𝐺𝐾 is equiangular and FG = 15, then what is the length of GK?