10. Proves the conditions for similarity of triangles.M9GE-Ilg-h-1MELC 11. Applies the theorems to show that given trianglesare similar. (M9GE-III-1)12. Proves the Pythagorean Theorem. (M9GE-III-2)As a continuation of our lessons last week, you will be proving andapplying the following theorems involving:10.5 Right Triangle Similarity Theorem10.6 Special Right Triangle TheoremsTRIANGLE SIMILARITYTriangle Angle-Bisector TheoremIf a segment bisects an angle of a triangle, then it divides the opposite sideinto segments proportional to the other two sides.IllustrationHD bisects LAHE,DAThen:Then: DEEHNotice: that sides on the numeratorsare adjacent. The same is true with thedenominators.ProofProveDAAHDEEHProofExtend AH to P such thatEP | HD.GivenA.HD bisects 4AHE.Exercise A. Complete the Proof table to prove the theorem.HintsStatementsReasonsNo.List down the givenDefinition of angleWhat happens to the21 =bisectorbisected LAHE?2| PageMathematics 92 HD | EPWhat can you say about3HD and EP ?By.What can you concludeabout ZADH & ZDEP andCorresponding anglesare congruent.21 & 24?Alternate interior anglesWhat can you concludeabout 22 & 43?are congruent.What can you sayabout 23 & z4?Transitive Propertybased on statements 2,4, and 5?Base angles of isoscelestriangles are congruent.What kind of triangle isДНЕР IsHEP based on statement6?What can you say aboutthe sides opposite 24? &23?Definition of isoscelestrianglesWhat can you sayabout ΔΑHD &ΔΑΡΕusing statement 4?AA Similarity TheoremUsing statement 3,10 write the preportionallengths of "APE.AHDefinition of SimilarAEPolygonsUse Segment Addition11ADSegment AdditionAH%3DPostulate for AP and AE.AH + AP AD +DEPostulateUse Inversion PropertyInversion Property of12 of Proportion statementProportion11.AH HP AD DEAH AHDecompose the fractions13 in statement 12 andsimplify.%3DAD AD Principles in theHPDE operations of fractionsAH-14 Simplify statement 13.Subtraction Property ofEqualityMathematics 93| Page5.

Here we have, In HD is ang;e bisector of ∠AHE⇒∠1=∠2 Prove:- DADE=AHEH

Answered: RIANGLE SIMILARITY Triangle…

RIANGLE SIMILARITY Triangle Angle-Bisector Theorem If a segment bisects an angle of a triangle, then it divides the opposite sid into segments proportional to the other two sides.

Trigonometry (MindTap Course List)

8th Edition

ISBN:9781305652224

Author:Charles P. McKeague, Mark D. Turner

Publisher:Charles P. McKeague, Mark D. Turner

Chapter1: The Six Trigonometric Functions

Section: Chapter Questions

Problem 2GP

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Question

10. Proves the conditions for similarity of triangles.
M9GE-Ilg-h-1
MELC 11. Applies the theorems to show that given triangles
are similar. (M9GE-III-1)
12. Proves the Pythagorean Theorem. (M9GE-III-2)
As a continuation of our lessons last week, you will be proving and
applying the following theorems involving:
10.5 Right Triangle Similarity Theorem
10.6 Special Right Triangle Theorems
TRIANGLE SIMILARITY
Triangle Angle-Bisector Theorem
If a segment bisects an angle of a triangle, then it divides the opposite side
into segments proportional to the other two sides.
Illustration
HD bisects LAHE,
DA
Then:
Then: DE
EH
Notice: that sides on the numerators
are adjacent. The same is true with the
denominators.
Proof
Prove
DA
AH
DE
EH
Proof
Extend AH to P such that
EP | HD.
Given
A.
HD bisects 4AHE.
Exercise A. Complete the Proof table to prove the theorem.
Hints
Statements
Reasons
No.
List down the given
Definition of angle
What happens to the
21 =
bisector
bisected LAHE?
2| Page
Mathematics 9
2

HD | EP
What can you say about
3
HD and EP ?
By.
What can you conclude
about ZADH & ZDEP and
Corresponding angles
are congruent.
21 & 24?
Alternate interior angles
What can you conclude
about 22 & 43?
are congruent.
What can you say
about 23 & z4?
Transitive Property
based on statements 2,
4, and 5?
Base angles of isosceles
triangles are congruent.
What kind of triangle is
ДНЕР Is
HEP based on statement
6?
What can you say about
the sides opposite 24? &
23?
Definition of isosceles
triangles
What can you say
about ΔΑHD &ΔΑΡΕ
using statement 4?
AA Similarity Theorem
Using statement 3,
10 write the preportional
lengths of "APE.
AH
Definition of Similar
AE
Polygons
Use Segment Addition
11
AD
Segment Addition
AH
%3D
Postulate for AP and AE.
AH + AP AD +DE
Postulate
Use Inversion Property
Inversion Property of
12 of Proportion statement
Proportion
11.
AH HP AD DE
AH AH
Decompose the fractions
13 in statement 12 and
simplify.
%3D
AD AD Principles in the
HP
DE operations of fractions
AH
-
14 Simplify statement 13.
Subtraction Property of
Equality
Mathematics 9
3| Page
5.

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