 # Using Properties of Angle Bisectors

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Using Properties of Angle Bisectors
Remember? The distance from a point to a line is defined as the length of the perpendicular segment from the point to the line. For instance, in the diagram shown, the distance between the point Q and the line m is QP.

10 minutes Rotation Reflection

If at first you don’t succeed, try and try again.
Geometry IB –HR Date: 2/13/ ID Check 2nd,4th, 6th, 7th Objective: SWBAT identify and use perpendicular and angle bisectors in triangles. Bell Ringer: 5 minute check 4.6/ minutes HW Requests: pg 304 #7-18/ Quadratics WS 2nd HW: pg Pg 327 #9-14, 21-26, 41, 42 Announcements: Quiz Section Thursday If at first you don’t succeed, try and try again.

Geometry IB_HR Date: 1/29/2014 ID Check
Objective: SWBAT identify and use perpendicular and angle bisectors in triangles. Bell Ringer: Turn In Take Home Test Due upon entry Bronson is creating a rt. triangular flower bed. If 2 sides of the flower bed are 7 ft long each, what is the length of the 3rd side to the nearest foot. Find the measure of each angle? HW Requests: None HW: pg 327 #9-20 Read Section 5.1 Announcements: Construction WS Due Friday 1/31 Life Is Just A Minute Life is just a minute—only sixty seconds in it. Forced upon you—can't refuse it. Didn't seek it—didn't choose it. But it's up to you to use it. You must suffer if you lose it. Give an account if you abuse it. Just a tiny, little minute, But eternity is in it! By Dr. Benjamin Elijah Mays, Past President of Morehouse College

Perpendicular Bisector – A segment, ray, line, or plane that is
perpendicular to a segment at its midpoint is called a perpendicular bisector. Perpendicular Bisectors in a triangle

Pg 322

Ex. 1 Using Perpendicular Bisectors
In the diagram MN is the perpendicular bisector of ST. What segment lengths in the diagram are equal? Explain why Q is on MN. c. If TM = 2x+3 and SM = 4x-7. What is the length of TM and SM?

Ex. 1 Using Perpendicular Bisectors
What segment lengths in the diagram are equal? Solution: MN bisects ST, so NS = NT. Because M is on the perpendicular bisector of ST, MS = MT. (By Theorem 5.1). The diagram shows that QS = QT = 12. Explain why Q is on MN. Solution: QS = QT, so Q is equidistant from S and T. By Theorem 5.2, Q is on the perpendicular bisector of ST, which is MN.

Perpendicular Bisector
Line, segment or ray that passes through the midpoint of the side and is perpendicular to that side. Circumcenter – intersection of the 3 bisectors. The circumcenter is equidistant from the vertices. If O is the circumcenter OA1 = OA2 = OA3.

Concurrent Lines: three or more lines intersect at a common point.
Point of concurrency: point where concurrent lines intersect.

Exit Ticket: pg 327 #1-4

Geometry IB_HR Date: 1/30/2014 ID Check
Objective: SWBAT identify and use perpendicular and angle bisectors in triangles. Bell Ringer: Get Triangle paper, Compass 4 paper clips Protractor, Ruler, 2 Pencils HW Requests: pg 327 #9-20 HW: pg 328 #21-29 odds, 32-35, 37, 41, 42, 45 Read Section 5.2 Announcements: Credit Recovery Registration Construction WS Due Friday 1/31 Life Is Just A Minute Life is just a minute—only sixty seconds in it. Forced upon you—can't refuse it. Didn't seek it—didn't choose it. But it's up to you to use it. You must suffer if you lose it. Give an account if you abuse it. Just a tiny, little minute, But eternity is in it! By Dr. Benjamin Elijah Mays, Past President of Morehouse College

Pg 325

On your paper label vertices of the triangle: A, B, C
Name: _____________________________________Date:________________Per:_____________ Constructions pg 321 Materials: Triangle paper Compass 4 paper clips Protractor Straightedge 2 Pencils Get your paper and tape it to your desk using 2 pieces of tape. Get 3 paper clips. On your paper label vertices of the triangle: A, B, C Using a compass (paper clips) and straightedge, construct a ⊥ bisector, PQ, on one side of the triangle. See directions below. Label the midpoint, M. Draw lines PA and PB (Answer Question 1) 6. Construct a ⊥ bisector on the 2 remaining sides of the triangle. Label each midpoint: M1, M2, M3 7. Label the intersection of the 3 ⊥ bisectors as point O. 8. Draw a line from each vertex, to the point O. (Answer Question 6)

Name: ________________________________
Date:___________________Per.______ Directions: See the Overhead Answer: the following Use complete sentences 1. Measures: AM = PA = BM = PB = 2. Use a protractor, what is the measure of <PMA and <PMB? 3. Is PQ a perpendicular bisector? Explain. 4. What is the name of the intersection of the 3 ⊥ bisectors? 5. Define – point of concurrency. Is the point found in #3 a point of concurrency? Explain, why. 6. Measures: OA = OB = OC = 7. What should be their relationship for line segments: OA, OB and OC? Reflection: What did you learn today that you did not already know?

Name: ________________________________
Date:___________________Per.______ Directions: See the Overhead Answer: the following Use complete sentences 1. Measures: AM = PA = BM = PB = 2. Use a protractor, what is the measure of <PMA and <PMB? 3. Is PQ a perpendicular bisector? Explain. 4. What is the name of the intersection of the 3 ⊥ bisectors? 5. Define – point of concurrency. Is the point found in #3 a point of concurrency? Explain, why. 6. Measures: OA = OB = OC = 7. What should be their relationship for line segments: OA, OB and OC? Reflection: What did you learn today that you did not already know?

Ex. 3: Using Angle Bisectors
Roof Trusses: Some roofs are built with wooden trusses that are assembled in a factory and shipped to the building site. In the diagram of the roof trusses shown, you are given that AB bisects CAD and that ACB and ADB are right angles. What can you say about BC and BD?

SOLUTION: Because BC and BD meet AC and AD at right angles, they are perpendicular segments to the sides of CAD. This implies that their lengths represent distances from the point B to AC and AD. Because point B is on the bisector of CAD, it is equidistant from the sides of the angle. So, BC = BD, and you can conclude that BC ≅ BD.

Theorem 5.1 Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If CP is the perpendicular bisector of AB, then CA = CB.

Theorem 5.2: Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If DA = DB, then D lies on the perpendicular bisector of AB.

What is the best way to track the constellations?
How does GPS work?

Placing Triangles on coordinate plane Key Concept pg 301
Step 1: Use the origin as a vertex or center of the triangle Step 2: Place at least one side of a triangle on an axis. Step 3: Keep the triangle within the first quadrant, if possible. Step 4: Use coordinates that make computations as simple as possible.

If at first you don’t succeed, try and try again.
Geometry HR Date: 2/8/ ID Check Objective: Identify reflections, translations, an rotations and verify congruence after a congruence transformation. Bell Ringer: See overhead HW Requests: pg 297 #7-23 odds Parking Lot: Perfect Square Trinomials, OEA #33 In class: Graph pg 298 #17-20, HW: Quadratic WS (Half Sheet) Announcements: Quiz Section Monday If at first you don’t succeed, try and try again.

If at first you don’t succeed, try and try again.
Geometry IB-HR Date: 2/7/ ID Check Objective: Identify reflections, translations, an rotations and verify congruence after a congruence transformation. Bell Ringer: Go over Red WB Sect. 4.6 HW Requests: pg 287 #9-21 odds, 29-32, 38, OEA #33 Parking Lot: Perfect Square Trinomials In class: Take Cornell Notes Pg 297 #1-6, pg 299 #24-26, 32 HW: pg 297 #7-23 odds Exit Ticket: pg 299 #24-26, 32 Announcements: Quiz Section Monday If at first you don’t succeed, try and try again.

Pg 294

If at first you don’t succeed, try and try again.
Geometry IB -HR Date: 2/4/ ID Check Objective: Use properties of isosceles and equilateral triangles. Bell Ringer: Put OEA in Bin - Go over OEA #46. HW Requests: Pg 291 #52-55 In class: Take Cornell Notes HW: pg 287 #9-21 odds, 29-32, 38, OEA #33; Read Sect. 4.7 Announcements: Quiz Section Monday If at first you don’t succeed, try and try again. Exit Ticket: Selected Problems pg 287 #1-7

Properties of Isosceles Triangles
Vertex Angle The angle formed by the congruent sides. Base Angle Two angles formed by the base and one of the congruent sides.

Thm. 4.10 -Isosceles Triangle Thm.
Ex: Proof 1 If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Thm. 4.11 Converse of Isosceles Triangle Theorem
Ex: Proof If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Equilateral Triangles
Corrollary 4.3 A  is equilateral if and only if it is equiangular. Corrollary 4.4 Each angle of an equilateral  measures 60 degrees.