Chapter 5 Section 2 Right Triangles.

Slides:

Advertisements

Similar presentations

Ways to Prove Triangles Congruent

Advertisements

Proving RightTriangles Congruent Free powerpoints at

CHAPTER 4 Congruent Triangles SECTION 4-1 Congruent Figures.

Chapter 4. Congruent Figures – figures that have exactly the same size and shape.Congruent Figures – figures that have exactly the same size and shape.

FINAL EXAM REVIEW Chapter 4 Key Concepts. Chapter 4 Vocabulary congruent figures corresponding parts equiangular Isosceles Δ legsbase vertex angle base.

3.7 Midsegments of Triangles

Chapter 5 Applying Congruent Triangles

Notes Lesson 5.2 Congruent Triangles Target 4.1.

4-7 Median, Altitude, and Perpendicular bisectors.

Medians, Altitudes and Perpendicular Bisectors

CHAPTER 4: CONGRUENT TRIANGLES

HW REVIEW Packet 4.7 (The rest of the Do Now Sheet)- Typo on number 6: change BC to AC and AD to BD Pg all Pg all and #18.

Chapter 4 Test Review.

& 5.2: Proving Triangles Congruent

Special Segments in Triangles Perpendicular bisector: A line or line segment that passes through the midpoint of a side of a triangle and is perpendicular.

Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

4.6 Congruence in Right Triangles

C. N. Colon Geometry St. Barnabas HS. Introduction Isosceles triangles can be seen throughout our daily lives in structures, supports, architectural details,

More About Triangles § 6.1 Medians

4.6 The Isosceles Triangle Theorems Base Angles and Opposite Sides Hypotenuse - Leg.

4.5 - Isosceles and Equilateral Triangles. Isosceles Triangles The congruent sides of an isosceles triangles are called it legs. The third side is the.

4-9 Isosceles and Equilateral Triangles

5-1 Special Segments in Triangles. I. Triangles have four types of special segments:

Chapter 5 Properties of Triangles Problems Chapter 5 Properties of Triangles Problems.

Chapter 7: Proportions and Similarity

6-4: Isosceles Triangles

Lesson 7.1 Right Triangles pp

Applying Congruent Triangles “Six Steps To Success”

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

Triangle Inequality Right, Acute, or Obtuse Isosceles Triangles Medians, Altitudes, & Bisectors $100 $200 $300 $400 $500.

 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 Special Segments in Triangles Learn about Perpendicular Bisector Learn about Medians Learn about Altitude Learn about Angle Bisector.

Isosceles and Equilateral Triangles Academic Geometry.

Isosceles Triangles Geometry Ms. Reed Unit 4, Day 2.

GEOMETRY HELP One student wrote “ CPA MPA by SAS” for the diagram below. Is the student correct? Explain. There are two pairs of congruent sides and one.

SPECIAL SEGMENTS IN TRIANGLES KEYSTONE GEOMETRY. 2 SPECIAL SEGMENTS OF A TRIANGLE: MEDIAN Definition of a Median: A segment from the vertex of the triangle.

Chapter 4 Ms. Cuervo. Vocabulary: Congruent -Two figures that have the same size and shape. -Two triangles are congruent if and only if their vertices.

Applied Geometry Lesson: 6 – 4 Isosceles Triangles Objective: Learn to identify and use properties of isosceles triangles.

A triangle in which exactly one angle is obtuse is called an ___________ triangle.

Side-side-side (SSS) postulate If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

How to use triangle congruence and CPCTC to prove that parts of two triangles are congruent, and HL to prove two triangles congruent. Chapter 4.4 and 4.6GeometryStandard/Goal:

5.4 Midsegment Theorem Geometry 2011.

The Isosceles Triangle Theorems

Triangles and Coordinate Proof

Congruence in Right Triangles

Triangle Fundamentals

Congruent Triangles TEST REVIEW

Triangle Fundamentals

4.2 APPLY CONGRUENCE AND TRIANGLES

Chapter 4 Section 4.1 – Part 1 Triangles and Angles.

4/20 Triangle Midsegment.

Triangle Fundamentals

Congruent Triangles 4-1: Classifying Triangles

4.5 - Isosceles and Equilateral Triangles

The Isosceles Triangle Theorems

4.1 Congruent Figures -Congruent Polygons: have corresponding angles and sides -Theorem 4.1: If 2 angles of 1 triangle are congruent to 2 angles of another.

Triangle Fundamentals

MID-TERM STUFF HONORS GEOMETRY.

5.3 Medians and Altitudes of a Triangle

EXAMPLE 1 Use similarity statements In the diagram, ∆RST ~ ∆XYZ

4-6 Congruence in Right Triangles

Lesson: 5.1 Special Segments in Triangles Pages: 238 – 241 Objectives:

(AAS) Angle-Angle-Side Congruence Theorem

Naming Triangles Triangles are named by using its vertices.

Midpoint and Median P9. The midpoint of the hypotenuse of a right triangle is equidistant from the 3 vertices. P12. The length of a leg of a right triangle.

5-2 Right Triangles Objectives:

Five-Minute Check (over Lesson 4–6) Mathematical Practices Then/Now

4/20 Triangle Midsegment.

4.4 The Isosceles Triangle Theorems Objectives: Legs/base Isosceles Triangle Th.

Presentation transcript:

Chapter 5 Section 2 Right Triangles

Warm-Up Draw and label a figure to illustrate each situation. 1) SL is an altitude of triangle RST and a perpendicular bisector of side RT. 2) Triangle HIJ is an isosceles triangle with vertex angle at J. JN is an altitude of triangle HIJ. State whether each sentence is always, sometimes, or never true. 3) The perpendicular bisectors of a triangle intersect at more than on point. 4) The altitude of a triangle contains the midpoint of the opposite side. 5) The medians of a triangle contain the midpoints of the opposite sides.

Warm-Up Draw and label a figure to illustrate each situation. 1) SL is an altitude of triangle RST and a perpendicular bisector of side RT. 2) Triangle HIJ is an isosceles triangle with vertex angle at J. JN is an altitude of triangle HIJ. S T R L J H I N

Warm-Up State whether each sentence is always, sometimes, or never true. 3) The perpendicular bisectors of a triangle intersect at more than on point. Never 4) The altitude of a triangle contains the midpoint of the opposite side. Sometimes 5) The medians of a triangle contain the midpoints of the opposite sides. Always

Vocabulary LL Theorem- If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. HA Theorem- If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.

Vocabulary cont. LA Theorem– If one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent. HL Postulate- If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.

<B and <D are right angles Example 1) State the additional information needed to prove each pair of triangles congruent by the given theorem or postulate. A) HL B) LL R S A D T B C U <B and <D are right angles V ST is congruent to TU C) LA L Q LN is congruent to QR or NM is congruent to RP P M N R

Example 2) Find the values of x and y so that triangle MNO is congruent to triangle RST. Assume triangle MNO is congruent to triangle RST. Then <N is congruent to <R and MO is congruent to RT. m<N = m<R 58 = 3y – 20 78 = 3y 26 = y MO = RT 47 – 8x = 15 -8x = -32 x = 4 58 O (3y – 20) (47 – 8x) cm M 15 cm T

Assume triangle JKL is congruent to triangle MLK. Example 3) Find the values of x and y so that triangle JKL is congruent to triangle MLK. Assume triangle JKL is congruent to triangle MLK. m<JLK = m<LKM 8y = 10y – 8 -2y = -8 y = 4 JK = LM 3 – 10x = 2x - 15 3 - 12x = -15 -12x = -18 x = 3/2 or 1.5 J (3 – 10x) (8y) L K (10y - 8) (2x - 15) M

A) AB = 2x + 6, BC = 15, AC = 3x + 4, XY = 20, YZ = x + 8; LL Example 4) Find the value of x so that triangle ABC is congruent to triangle XYZ by the indicated theorem or postulate. A) AB = 2x + 6, BC = 15, AC = 3x + 4, XY = 20, YZ = x + 8; LL Let XY be congruent to AB and YZ be congruent to BC. XY = AB 20 = 2x + 6 14 = 2x 7 = x YZ = BC x + 8 = 15 x = 7 So x is equal to 7. A Y Z B C X

B) m<Z = 55, BC = 15x + 2, m<C = 55, AB = 24, YZ = 4x + 13; LA Example 4) Find the value of x so that triangle ABC is congruent to triangle XYZ by the indicated theorem or postulate. B) m<Z = 55, BC = 15x + 2, m<C = 55, AB = 24, YZ = 4x + 13; LA Let <Z be congruent to <C and YZ be congruent to BC. YZ = BC 4x + 13 = 15x + 2 13 = 11x + 2 11 = 11x 1 = x So x is equal to 1. A Y Z B C X

C) YX = 21, m<X = 9x + 9, AB = 21, m<A = 11x - 3; LA Example 4) Find the value of x so that triangle ABC is congruent to triangle XYZ by the indicated theorem or postulate. C) YX = 21, m<X = 9x + 9, AB = 21, m<A = 11x - 3; LA Let XY be congruent to AB and <X be congruent to <A. m<X = m<A 9x + 9 = 11x - 3 9 = 2x – 3 12 = 2x 6 = x So x is equal to 6. A Y Z B C X

D) AC = 28, AB = 7x + 4, ZX = 9x + 1, YX = 5(x + 2); HL Example 4) Find the value of x so that triangle ABC is congruent to triangle XYZ by the indicated theorem or postulate. D) AC = 28, AB = 7x + 4, ZX = 9x + 1, YX = 5(x + 2); HL Let AC be congruent to ZX and AB be congruent to YX. XY = AB 5(x + 2) = 7x + 4 5x + 10 = 7x + 4 10 = 2x + 4 6 = 2x 3 = x AC = ZX 28 = 9x + 1 27 = 9x So x is equal to 3. A Y Z B C X