Section 5-2 Bisectors in Triangles. Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

Slides:

Advertisements

Similar presentations

Bisectors in Triangles Academic Geometry. Perpendicular Bisectors and Angle Bisectors In the diagram below CD is the perpendicular bisector of AB. CD.

Advertisements

Section 4-3 Triangle Congruence (ASA, AAS) SPI 32C: determine congruence or similarity between triangles SPI 32M: justify triangle congruence given a diagram.

4.6 Using Congruent Triangles

Chapter 9 Geometry © 2008 Pearson Addison-Wesley. All rights reserved.

Section 9-3 The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem.

Geo: Sec 5.1 Perpendiculars & Bisectors Thm 5.1 Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant.

5.1 Perpendiculars and Bisectors Geometry Mrs. Spitz Fall 2004.

5.1 Perpendiculars and Bisectors Geometry Mrs. Spitz Fall 2004.

Benchmark 24 I can use properties of perpendicular bisectors and angle bisectors to identify equal distances.

5-2 Perpendicular and Angle Bisectors Learning Goals 1. To use properties of perpendicular bisectors and angle bisectors.

4.1 Detours & Midpoints Obj: Use detours in proofs Apply the midpoint formulas Apply the midpoint formulas.

What are the ways we can prove triangles congruent? A B C D Angle C is congruent to angle A Angle ADB is congruent to angle CDB BD is congruent to BD A.

Similar Triangle Proofs Page 5-7. A CB HF E Similar Triangle Proof Notes To prove two triangles are similar, you only need to prove that 2 corresponding.

EXAMPLE 4 Use the Third Angles Theorem Find m BDC. So, m ACD = m BDC = 105° by the definition of congruent angles. ANSWER SOLUTION A B and ADC BCD, so.

EXAMPLE 4 Use the Third Angles Theorem Find m BDC. So, m ACD = m BDC = 105 ° by the definition of congruent angles. ANSWER SOLUTION A B and ADC BCD, so.

Isosceles Triangles Geometry D – Chapter 4.6. Definitions - Review Define an isosceles triangle. A triangle with two congruent sides. Name the parts of.

Stuck on 4.1 – 4.4? Katalina Urrea and Maddie Stein ;)

Isosceles, Equilateral, and Right Triangles Geometry Mrs. Kinser Fall 2012.

5.1 Perpendiculars and Bisectors Geometry Mrs. Spitz Fall 2004.

Warm Up Week 7 Label the information: AB C D 1) ∠C is a right angle. 2) ∠A ≅ ∠B 3) AB = 8 4) bisects.

Given: DB bisects AE

EXAMPLE 3 Write a flow proof In the diagram, CE BD and  CAB CAD. Write a flow proof to show ABE ADE GIVEN CE BD,  CAB CAD PROVE ABE ADE.

Identify the Property which supports each Conclusion.

Other methods of Proving Triangles Congruent (AAS), (HL)

(4.4) CPCTC Corresponding Parts of Congruent Triangles Congruent

Math Section 4.4 Equidistance Theorems By: Dan Schlosser and Brad Wagner.

EXAMPLE 1 Identify congruent triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem.

4.6 Prove Triangles Congruent by ASA and AAS

5.1 midsegments of triangles Geometry Mrs. Spitz Fall 2004.

Perpendicular Bisectors ADB C CD is a perpendicular bisector of AB Theorem 5-2: Perpendicular Bisector Theorem: If a point is on a perpendicular bisector.

Section 5-1 Perpendiculars and Bisectors. Perpendicular bisector A segment, ray, line, or plane that is perpendicular to a segment at its midpoint.

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

 In Chapter 1, you learned the definition of a midpoint of a segment. What do you think a midsegment of a triangle is?  Find the midpoint of AB: o A(-2,

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

AAS examples By: Ana Cristina Andrade. A D C E V V Given: segment AD is parallel to segment BC. Segment AD is congruent to segment CB Proof: Triangle.

Holt McDougal Geometry 4-Ext Proving Constructions Valid 4-Ext Proving Constructions Valid Holt Geometry Lesson Presentation Lesson Presentation Holt McDougal.

Section 5.2 Use Perpendicular Bisectors. Vocabulary Perpendicular Bisector: A segment, ray, line, or plane that is perpendicular to a segment at its midpoint.

EXAMPLE 3 Write a flow proof In the diagram, CE BD and  CAB CAD. Write a flow proof to show ABE ADE GIVEN CE BD,  CAB CAD PROVE ABE ADE.

Isosceles and Equilateral Triangles

4.4 Isosceles Triangles, Corollaries, & CPCTC. ♥Has at least 2 congruent sides. ♥The angles opposite the congruent sides are congruent ♥Converse is also.

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.

By Shelby Smith and Nellie Diaz. Section 8-1 SSS and SAS  If three sides of one triangle are congruent to three sides of another triangle, then the triangles.

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.

CPCTC  ’s Naming  ’s Algebra Connection Proofs Altitudes & Medians

LESSON TEN: CONGRUENCE CONUNDRUM. CONGRUENCE We have already discussed similarity in triangles. What things must two triangles have in order to be similar?

5.2: Bisectors in Triangles Objectives: To use properties of perpendicular and angle bisectors.

5.1 Perpendiculars and Bisectors Geometry Mrs. Spitz Fall 2004.

EXAMPLE 3 Find the orthocenter Find the orthocenter P in an acute, a right, and an obtuse triangle. SOLUTION Acute triangle P is inside triangle. Right.

Lesson 4-7: Medians, Altitudes, and Perpendicular Bisectors (page 152) Essential Questions Can you construct a proof using congruent triangles?

1. Prove that the three angle bisectors of a triangle concur. C AB D F E I § 4.1.

Do Now: 1. Name how the two triangles are congruent in the rectangle below: 2. Find the measure of an exterior angle in a pentagon: Homework: Packet page.

4-4 Using Corresponding Parts of Congruent Triangles I can determine whether corresponding parts of triangles are congruent. I can write a two column proof.

Chapter 5: Properties of Triangles Section 5.1: Perpendiculars and Bisectors.

LESSON 5-2 BISECTORS IN TRIANGLES OBJECTIVE:

LESSON 5-3 MEDIANS, ALTITUDES & ANGLE BISECTORS OF TRIANGLES

5.1 Perpendiculars and Bisectors

1. When are two angles congruent?

5.1 Perpendiculars and Bisectors

5.2 Bisectors in Triangles

A segment that connects a vertex of a triangle

Prove Triangles Congruent by ASA & AAS

Check your answers from 5.2 (p )

P ;#10, all, 28, 31, 33, 35, 36-40(e), ) JK = √10, KL = √10, JL = √20, FG = √10, GH = √10, FH = √20, not congruent b/c corresponding.

Geometry Proofs Unit 12 AA1.CC.

5.2 Bisectors in Triangles

Module 15: Lesson 5 Angle Bisectors of Triangles

Bell Work Complete problems 8, 9, and 15 from yesterday. Proofs are on the board.

Geometry/Trig Name: __________________________

Presentation transcript:

Section 5-2 Bisectors in Triangles

Vocabulary Distance from a point to a line: the length of the perpendicular segment from the point to the line.

Theorems Perpendicular Bisector Theorem- If a point lies on the perpendicular bisector of a line segment, then it is an equal distance away from both endpoints of the line segment. Angle Bisector Theorem- If a point lies on the angle bisector of an angle, then it is an equal distance away from both sides of the angle.

Converse of the Theorems Converse of the Perpendicular Bisector Theorem- If a point is an equal distance away from the endpoints of a line segment, then it lies on the perpendicular bisector of the line segment. Converse of the Angle Bisector Theorem- If a point in the interior of an angle is an equal distance away from both sides of the angle, then it lies on the angle bisector of the angle.

Perpendicular Bisector Theorem

Angle Bisector Theorem

Proof of Perpendicular Bisector Theorem StatementReason BD is the ┴ bisector of AC Given ∠ABD and ∠CBD are right angles Definition of perpendicular ∠ABD ≅ ∠CBD All right angles are congruent DB ≅ DB Reflexive Property of Congruency AB ≅ CB Definition of bisector ∆ABD ≅ ∆CBD SAS AD ≅ CD CPCTC

Proof of Angle Bisector Theorem StatementReason AD is the angle bisector of ∠CAB Given CD is ┴ to AC By construction DB is ┴ to AB By construction ∠ACB and ∠ABD are right angles Definition of perpendicular ∠ACB ≅ ∠ABD All right angles are congruent ∠CAD ≅ ∠BAD Definition of angle bisector AD ≅ AD Reflexive Property pf Congruency ∆CAD ≅ ∆BAD AAS CD ≅ BD CPCTC

Practice Problem Given: BE is the perpendicular bisector of AC, ∠AED ≅ ∠CEF, DE ≅ FE. Prove: ∠DAE ≅ ∠FCE Answer on next slide⫸

Solution to Practice Problem StatementReason BE is ┴ bisector of ACGiven AE ≅ CE Perpendicular Bisector Theorem DE ≅ FE Given ∠AED ≅ ∠CEF Given ∆ADE ≅ ∆CFE SAS ∠DAE ≅ ∠FCE CPCTC

Practice Problem 2 Find the value of X and Y Answer on next slide⫸

Solution to Practice Problem 2 Answer: X = -3, Y = 12

Extra Resources (very weird jumpy guy explaining the perpendicular bisector theorem) 8AB35C6885A036&index=35 (same weird guy explaining the angle bisector theorem) 8AB35C6885A036&index=35