Warm Up 1. Write the angles in order from smallest to largest.
-
Upload
janna-ryan -
Category
Documents
-
view
35 -
download
0
description
Transcript of Warm Up 1. Write the angles in order from smallest to largest.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Warm Up
1. Write the angles in order from smallest to largest.
2. The lengths of two sides of a triangle are 12 cm and 9 cm. Find the range of possible lengths for the third side.
X, Z, Y
3 cm < s < 21 cm
Holt McDougal Geometry
5-6 Inequalities in Two Triangles5-6 Inequalities in Two Triangles
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt McDougal Geometry
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Apply inequalities in two triangles.
Objective
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Example 1A: Using the Hinge Theorem and Its Converse
Compare mBAC and mDAC.
Compare the side lengths in ∆ABC and ∆ADC.
By the Converse of the Hinge Theorem, mBAC > mDAC.
AB = AD AC = AC BC > DC
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Example 1B: Using the Hinge Theorem and Its Converse
Compare EF and FG.
By the Hinge Theorem, EF < GF.
Compare the sides and angles in ∆EFH angles in ∆GFH.
EH = GH FH = FH mEHF > mGHF
mGHF = 180° – 82° = 98°
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Example 1C: Using the Hinge Theorem and Its Converse
Find the range of values for k.
Step 1 Compare the side lengths in ∆MLN and ∆PLN.
By the Converse of the Hinge Theorem, mMLN > mPLN.
LN = LN LM = LP MN > PN
5k – 12 < 38
k < 10
Substitute the given values.
Add 12 to both sides and divide by 5.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Example 1C Continued
Step 2 Since PLN is in a triangle, mPLN > 0°.
Step 3 Combine the two inequalities.
The range of values for k is 2.4 < k < 10.
5k – 12 > 0
k < 2.4
Substitute the given values.
Add 12 to both sides and divide by 5.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Check It Out! Example 1a
Compare mEGH and mEGF.
Compare the side lengths in ∆EGH and ∆EGF.
FG = HG EG = EG EF > EH
By the Converse of the Hinge Theorem, mEGH < mEGF.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Check It Out! Example 1b
Compare BC and AB.
Compare the side lengths in ∆ABD and ∆CBD.
By the Hinge Theorem, BC > AB.
AD = DC BD = BD mADB > mBDC.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Example 2: Travel Application
John and Luke leave school at the same time. John rides his bike 3 blocks west and then 4 blocks north. Luke rides 4 blocks east and then 3 blocks at a bearing of N 10º E. Who is farther from school? Explain.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Example 2 Continued
The distances of 3 blocks and 4 blocks are the same in both triangles.
The angle formed by John’s route (90º) is smaller than the angle formed by Luke’s route (100º). So Luke is farther from school than John by the Hinge Theorem.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Check It Out! Example 2
When the swing ride is at full speed, the chairs are farthest from the base of the swing tower. What can you conclude about the angles of the swings at full speed versus low speed? Explain.
The of the swing at full speed is greater than the at low speed because the length of the triangle on the opposite side is the greatest at full swing.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Example 3: Proving Triangle Relationships
Write a two-column proof.
Given:
Prove: AD > CB
Proof:
Statements Reasons
1. Given
2. Reflex. Prop. of
3. Hinge Thm.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Check It Out! Example 3a
Write a two-column proof.
Given: C is the midpoint of BD.
Prove: AB > ED
m1 = m2
m3 > m4
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
1. Given
2. Def. of Midpoint
3. Def. of s
4. Conv. of Isoc. ∆ Thm.
5. Hinge Thm.
1. C is the mdpt. of BDm3 > m4, m1 = m2
3. 1 2
5. AB > ED
Statements Reasons
Proof:
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Write a two-column proof.
Given:
Prove: mTSU > mRSU
Statements Reasons
1. Given
3. Reflex. Prop. of
4. Conv. of Hinge Thm.
2. Conv. of Isoc. Δ Thm.
1. SRT STRTU > RU
SRT STRTU > RU
Check It Out! Example 3b
4. mTSU > mRSU
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Lesson Quiz: Part I
1. Compare mABC and mDEF.
2. Compare PS and QR.
mABC > mDEF
PS < QR
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Lesson Quiz: Part II
3. Find the range of values for z.
–3 < z < 7