Chapter 5 Applying Congruent Triangles Warm Up For Chapter 5 5.1 Special Segments in Triangles 5.1 Day 2 Proofs 5.2 Right Triangles Internet Activity

5.1 Special Segments in Triangles Objective: Identify and use medians, altitudes, angle bisectors, and perpendicular bisectors in a triangle Click Me!! How will I use this? Special segments are used in triangles to solve problems involving engineering, sports and physics. Angle Bisector Chapter 5 Median Perpendicular Bisector Altitude An example to tie it all together

Definitions Median A segment that connects a vertex of a triangle to the midpoint of the side opposite the vertex.

Definitions Altitude A line segment with 1 endpoint at a vertex of a triangle and the other on the line opposite that vertex so that the line segment is perpendicular to the side of the triangle.

Perpendicular Bisector Theorems! Definitions Perpendicular Bisector: A line or line segment that passes through the midpoint of a side of a triangle and is perpendicular to that side. Perpendicular Bisector Theorems!

Perpendicular Bisector Theorems Theorem 5.1: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Theorem 5.2: Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. Angle Bisector Chapter 5 Median Perpendicular Bisector Altitude

Perpendicular Bisector Theorems Theorem 5.3: Any point on the bisector of an angle is equidistant from the sides of the angle. Theorem 5.4: Any point on or in the interior of an angle and equidistant from the sides of an angle lies on the bisector of the angle. Angle Bisector Chapter 5 Median Perpendicular Bisector Altitude

Warm UP In Find the value of x and the measure of each angle. Answer

What type of triangle is ABC? Click me to find the Answer!! Warm Up Answers How did I get that? Click the answer to see! BONUS!!! What type of triangle is ABC? Click me to find the Answer!! Section 5.1

What type of triangle is ABC? Click me to find the Answer!! BONUS!!! What type of triangle is ABC? Click me to find the Answer!!

Because the question give you angle measures, we take the sum of the angles and set them equal to 180. } } } BONUS!! Click Me!! Combine like terms! Add 20 to both sides! Divide by 10 on both sides! Section 5.1 Chapter 5

Use substitution for the answer you found for x and plug it into the equation for angle A. BONUS!! Click Me!! Section 5.1 Chapter 5

Use substitution for the answer you found for x and plug it into the equation for angle B. BONUS!! Click Me!! Section 5.1 Chapter 5

Use substitution for the answer you found for x and plug it into the equation for angle C. BONUS!! Click Me!! Section 5.1 Chapter 5

Why is that?? Triangle ABC is a right isosceles triangle Section 5.1 Chapter 5

Angle Bisector Move my vertices around and see what happens!! What is an Angle Bisector? Click me to find out! Move my vertices around and see what happens!! Angle Bisector Theorems Example Section 5.1

Draw the three medians of triangle ABC. Name each of them. Median Example Draw the three medians of triangle ABC. Name each of them. B Answer C A

Draw the three medians of triangle ABC. Name each of them. Median Example Draw the three medians of triangle ABC. Name each of them. B F D C A E Back to Section 5.1

Draw the three altitudes, QU, SV, and RT. Altitude Example Draw the three altitudes, QU, SV, and RT. Q R S Answer

Draw the three altitudes, QU, SV, and RT. Altitude Example Draw the three altitudes, QU, SV, and RT. V U Q R S T Back to Section 5.1

Perpendicular Bisector Example Draw the three lines that are perpendicular bisectors of XYZ. Y Answer X Z Label the lines l, m, and n.

Perpendicular Bisector Example Draw the three lines that are perpendicular bisectors of XYZ. Y m l X n Z Back to Section 5.1

Angle Bisector Example If BD bisects ABC, find the value of x and the measure of AC. B A C D Answer

Angle Bisector Example If BD bisects ABC, find the value of x and the measure of AC. B Show me how you got those answers! A C D Back to Section 5.1

Angle Bisector Example If BD bisects ABC, find the value of x and the measure of AC. Means that the angle is split into 2 congruent parts. Set the two angles equal to each other and solve. B Once you find x, plug it into AD and DC. Since you are looking for the total length, AC, use segment addition to find the total length. A C D Back to Section 5.1

5.1 Proofs Together YOU TRY!!!

Given: Prove: 1. Given 5. SAS 2. Def of Isos Triangle 6. CPCTC Just keep clicking! Prove: 1 2 5 1 3 6 4 7 1. Given 5. SAS 2. Def of Isos Triangle 6. CPCTC 3. Def of Angle Bisector 7. Def of Median 5.1 Proofs 4. Reflexive

We’re done, take me back to the beginning! Given: Just keep clicking! Prove: 1 2 5 1 3 6 4 7 1. Given 5. SAS 2. Def of Equilateral Triangle 6. CPCTC We’re done, take me back to the beginning! 3. Def of Angle Bisector 7. Def of Median 4. Reflexive

Keep clicking to see graph! Example Keep clicking to see graph! S G B median Midpoint See the Work!!

What is the Midpoint Formula? Midpoint of GB Just keep clicking! Next Question

We’re done, take me back to the beginning! Just keep clicking! What can we conclude? We’re done, take me back to the beginning!

An Internet Activity CLICK TO BEGIN 5.2 Right Triangles An Internet Activity CLICK TO BEGIN

Click on the triangle and learn about the Theorems or Postulates. Take notes as you read along with each Theorem or Postulate!! Leg Leg Theorem Leg Angle Theorem Hypotenuse Angle Theorem Hypotenuse Leg Postulate Click me when done

Examples Solving for variables I finished! Click me!! Stating additional information

Solve for… Example 1 Example 2 Example 3

State the additional information. Example 1 2 3

D P F E Q R Answer

Next Example

E F D Q R P Answer

Next Example

F E P D Q R Answer

Back to Beginning

State the additional information needed to prove the pair of triangles congruent by LA. J K L Answer

Proving triangles congruent by LA means a leg and an angle of the right triangle must be congruent. J K L OR Next Example

State the additional information needed to prove the pair of triangles congruent by HA. S Z T X Y V Answer

State the additional information needed to prove the pair of triangles congruent by HA. T S Z Y X V The keyword was additional. When proving triangles congruent by HA, all that is needed is to show that the hypotenuse is congruent on each triangle as well as an acute angle. In these triangles both are already shown so there is no ADDITIONAL information needed. Next Example

State the additional information needed to prove the pair of triangles congruent by LA. Answer F

State the additional information needed to prove the pair of triangles congruent by LA. Back to Beginning