EXAMPLE 1 Use the Angle Bisector Theorems Find the measure of GFJ.

Slides:

Advertisements

Similar presentations

Congruent Triangles In two congruent figures, all the parts of one figure are congruent to the corresponding parts of the other figure. This means there.

Advertisements

Lesson 6.1 – Properties of Tangent Lines to a Circle

Chapter 5 – Properties of Triangles The Bigger Picture -Properties of Triangles such as perpendicular and angle bisectors and how they relate in triangles.

Chapter 5 Congruent Triangles. 5.1 Perpendiculars and Bisectors Perpendicular Bisector: segment, line, or ray that is perpendicular and cuts a figure.

5.1 Midsegment Theorem and Coordinate Proof Objectives: 1.To discover and use the Midsegment Theorem 2.To write a coordinate proof.

Points of Concurrency in Triangles Keystone Geometry

In the diagram of C, QR = ST = 16. Find CU.

Chapter 5 Angle Bisectors. Angle Bisector A ray that bisects an angle into two congruent angles.

5.3 Use Angle Bisectors of Triangles

EXAMPLE 1 Use the Angle Bisector Theorems SOLUTION Because JG FG and JH FH and JG = JH = 7, FJ bisects GFH by the Converse of the Angle Bisector Theorem.

Ch 5.3 Use Angle bisectors of triangles. In this section… We will use the properties of an angle bisector to solve for missing side lengths.

EXAMPLE 1 Find the length of a hypotenuse SOLUTION Find the length of the hypotenuse of the right triangle. (hypotenuse) 2 = (leg) 2 + (leg) 2 Pythagorean.

5.3Use Angle Bisectors of Triangles Theorem 5.5: Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two.

MM2G3 Students will understand properties of circles. MM2G3 d Justify measurements and relationships in circles using geometric and algebraic properties.

Section 11-2 Chords and Arcs SPI 32B: Identify chords of circles given a diagram SPI 33A: Solve problems involving the properties of arcs, tangents, chords.

5.2 Bisectors of a Triangle Geometry Ms. Reser. Objectives Use properties of perpendicular bisectors of a triangle as applied in Example 1. Use properties.

Bell Problem Find the value of x Use Angle Bisectors of Triangles Standards: 1.Represent situations using algebraic symbols 2.Analyze properties.

Section 5.1 Bisectors of Triangles. We learned earlier that a segment bisector is any line, segment, or plane that intersects a segment at its midpoint.

1. Solve x2 = 100. ANSWER 10, –10 2. Solve x2 + 9 = 25. ANSWER 4, –4

EXAMPLE 1 Use the Angle Bisector Theorems Find the measure of GFJ.

5.3 Using Angle Bisectors of Triangles. Vocabulary/Theorems Angle bisector: ray that divides angle into 2 congruent angles Point of concurrency: point.

Warm-Up Exercises 2. Solve x = 25. ANSWER 10, –10 ANSWER 4, –4 1. Solve x 2 = 100. ANSWER Simplify 20.

Points of Concurrency in Triangles (5-1 through 5 – 4)

5-1 Bisectors, Medians, and Altitudes 1.) Identify and use perpendicular bisectors and angle bisectors in triangles. 2.) Identify and use medians and altitudes.

MM2G3a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

5-1 Bisectors of Triangles

5.3 – Use Angle Bisectors of Triangles

Bisectors in Triangles Section 5-2. Perpendicular Bisector A perpendicular tells us two things – It creates a 90 angle with the segment it intersects.

Chapter 5.3 Notes: Use Angle Bisectors of Triangles Goal: You will use angle bisectors to find distance relationships.

5.1 Perpendiculars and Bisectors. Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from.

5.2 Bisectors of a Triangle Geometry Mr. Davenport Fall 2009.

10.3 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Apply Properties of Chords.

Daily Warm-Up Quiz F H, J, and K are midpoints 1. HJ = __ 60 H 65 J FG = __ JK = __ m < HEK = _ E G Reason: _ K 3. Name all 100 parallel segments.

Isosceles Triangles Theorems Theorem 8.12 – If two sides of a triangle are equal in measure, then the angles opposite those sides are equal in measure.

Perpendicular and Angle Bisectors Perpendicular Bisector – A line, segment, or ray that passes through the midpoint of a side of a triangle and is perpendicular.

EXAMPLE 4 Use the concurrency of angle bisectors In the diagram, N is the incenter of ABC. Find ND. SOLUTION By the Concurrency of Angle Bisectors of a.

Bisectors of a Triangle Geometry. Objectives Use properties of perpendicular bisectors of a triangle. Use properties of angle bisectors of a triangle.

5.3.1 Use Angle Bisectors of Triangles Chapter 5: Relationships within Triangles SWBAT: Define and use Angle Bisector Theorem. Define incenter You will.

5.3 Use Angle Bisectors of Triangles Hubarth Geometry.

Bisectors of a Triangle Geometry Objectives Use properties of angle bisectors of a triangle. Use properties of perpendicular bisectors of a triangle.

5.3 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Use Angle Bisectors of Triangles.

Get your journal What did one pencil say to the other pencil?

Section 5.2: Bisectors of a Triangle. Perpendicular bisector of a triangle – A line, ray, or segment that is perpendicular to a side of the triangle at.

Special Segments in Triangles

EXAMPLE 3 Find the area of an isosceles triangle

1. DC Tell whether the segment is best described as a radius,

LESSON 5-2 BISECTORS IN TRIANGLES OBJECTIVE:

1. AD bisects BAC. Find x. ANSWER 20.

Use Angle Bisectors of Triangles

1. AD bisects BAC. Find x. ANSWER Solve x² + 12² = 169. ANSWER

Table of Contents Date: Topic: Description: Page:.

Perpendicular Bisectors

1. DC Tell whether the segment is best described as a radius,

1. Solve 3x = 8x – 15. ANSWER 3 2. Solve 6x + 3 = 8x – 14. ANSWER 8.5.

Use the Perpendicular Bisector Theorem

5.2 Bisectors of a Triangle

EXAMPLE 1 Use the Angle Bisector Theorems Find the measure of GFJ.

Section 5.3 Use Angle Bisectors of Triangles

**Homework: #1-6 from 5.1 worksheet**

5.2 Bisectors of a Triangle

EXAMPLE 1 Use congruent chords to find an arc measure

EXAMPLE 1 Identify special segments and lines

Module 15: Lesson 5 Angle Bisectors of Triangles

Use Perpendicular Bisectors

1. Solve x2 = 100. ANSWER 10, –10 2. Solve x2 + 9 = 25. ANSWER 4, –4

Use Angle Bisectors of Triangles

Bisectors Concept 35.

5-1 Bisectors of Triangles

Objectives: I will use perpendicular bisector to solve Problems.

Presentation transcript:

EXAMPLE 1 Use the Angle Bisector Theorems Find the measure of GFJ. SOLUTION Because JG FG and JH FH and JG = JH = 7, FJ bisects GFH by the Converse of the Angle Bisector Theorem. So, m GFJ = m HFJ = 42°.

EXAMPLE 2 Solve a real-world problem SOCCER A soccer goalie’s position relative to the ball and goalposts forms congruent angles, as shown. Will the goalie have to move farther to block a shot toward the right goalpost R or the left goalpost L?

EXAMPLE 2 Solve a real-world problem SOLUTION The congruent angles tell you that the goalie is on the bisector of LBR. By the Angle Bisector Theorem, the goalie is equidistant from BR and BL . So, the goalie must move the same distance to block either shot.

Use algebra to solve a problem EXAMPLE 3 Use algebra to solve a problem ALGEBRA For what value of x does P lie on the bisector of A? SOLUTION From the Converse of the Angle Bisector Theorem, you know that P lies on the bisector of A if P is equidistant from the sides of A, so when BP = CP. BP = CP Set segment lengths equal. x + 3 = 2x –1 Substitute expressions for segment lengths. 4 = x Solve for x. Point P lies on the bisector of A when x = 4.

GUIDED PRACTICE for Examples 1, 2, and 3 In Exercises 1–3, find the value of x. 1. A B C P 15 ANSWER

GUIDED PRACTICE for Examples 1, 2, and 3 In Exercises 1–3, find the value of x. 2. A B C P 11 ANSWER

GUIDED PRACTICE for Examples 1, 2, and 3 In Exercises 1–3, find the value of x. 3. P B C A 5 ANSWER

GUIDED PRACTICE for Examples 1, 2, and 3 4. Do you have enough information to conclude that QS bisects PQR? Explain. No; you need to establish that SR QR and SP QP. ANSWER

EXAMPLE 4 Use the concurrency of angle bisectors In the diagram, N is the incenter of ABC. Find ND. SOLUTION By the Concurrency of Angle Bisectors of a Triangle Theorem, the incenter N is equidistant from the sides of ABC. So, to find ND, you can find NF in NAF. Use the Pythagorean Theorem.

Use the concurrency of angle bisectors EXAMPLE 4 Use the concurrency of angle bisectors c = 2 a + b Pythagorean Theorem NF + 16 2 20 = Substitute known values. 400 = NF + 256 2 Multiply. 144 = NF 2 Subtract 256 from each side. 12 = NF Take the positive square root of each side. Because NF = ND, ND = 12.

GUIDED PRACTICE for Example 4 5. WHAT IF? In Example 4, suppose you are not given AF or AN, but you are given that BF = 12 and BN = 13. Find ND. 5 ANSWER

Daily Homework Quiz Find the value of x. 1. ANSWER 12

Daily Homework Quiz 2. ANSWER The value of x cannot be determined because the congruent segments are not known to be perpendicular to the sides of the angle.

Daily Homework Quiz 3. Point D is the incenter of XYZ. Find DB. ANSWER 10