Class 9 Exam > Class 9 Questions > The bisectors of base angles q and R of an eq...
Start Learning for Free
The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr?
Most Upvoted Answer
The bisectors of base angles q and R of an equilateral triangle pqr in...
Community Answer
The bisectors of base angles q and R of an equilateral triangle pqr in...
Problem Statement:
In an equilateral triangle PQR, the bisectors of the base angles Q and R intersect at point S. ST and SM are drawn parallel to the sides PQ and PR, intersecting side QR at points T and M respectively. We need to prove that QT = CM = MR.
Proof:
Step 1: Draw the diagram
Let's start by drawing an equilateral triangle PQR with the bisectors of angles Q and R intersecting at point S. Then draw parallel lines ST and SM to sides PQ and PR respectively, intersecting side QR at points T and M.
Step 2: Identify the key points and angles
- Point S is the intersection of the bisectors of angles Q and R.
- Point T is the intersection of line ST with side QR.
- Point M is the intersection of line SM with side QR.
- We need to prove that QT = CM = MR.
Step 3: Identify the congruent triangles
To prove that QT = CM = MR, we can show that triangle QST is congruent to triangle MSR.
Step 4: Prove congruence of triangles QST and MSR
- QTS and SRT are congruent by the Angle-Bisector Theorem, as they share a common side ST and have equal angles at S.
- STR and STM are congruent by the Alternate Interior Angles Theorem, as they are corresponding angles formed by parallel lines.
- Therefore, triangles QST and MSR are congruent by the Side-Angle-Side (SAS) congruence criterion.
- Congruent triangles have equal corresponding sides, so QT = CM = MR.
Step 5: Conclusion
Therefore, we have proved that QT = CM = MR using congruence of triangles QST and MSR.
In an equilateral triangle PQR, the bisectors of the base angles Q and R intersect at point S. ST and SM are drawn parallel to the sides PQ and PR, intersecting side QR at points T and M respectively. We need to prove that QT = CM = MR.
Proof:
Step 1: Draw the diagram
Let's start by drawing an equilateral triangle PQR with the bisectors of angles Q and R intersecting at point S. Then draw parallel lines ST and SM to sides PQ and PR respectively, intersecting side QR at points T and M.
Step 2: Identify the key points and angles
- Point S is the intersection of the bisectors of angles Q and R.
- Point T is the intersection of line ST with side QR.
- Point M is the intersection of line SM with side QR.
- We need to prove that QT = CM = MR.
Step 3: Identify the congruent triangles
To prove that QT = CM = MR, we can show that triangle QST is congruent to triangle MSR.
Step 4: Prove congruence of triangles QST and MSR
- QTS and SRT are congruent by the Angle-Bisector Theorem, as they share a common side ST and have equal angles at S.
- STR and STM are congruent by the Alternate Interior Angles Theorem, as they are corresponding angles formed by parallel lines.
- Therefore, triangles QST and MSR are congruent by the Side-Angle-Side (SAS) congruence criterion.
- Congruent triangles have equal corresponding sides, so QT = CM = MR.
Step 5: Conclusion
Therefore, we have proved that QT = CM = MR using congruence of triangles QST and MSR.
Attention Class 9 Students!
To make sure you are not studying endlessly, EduRev has designed Class 9 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 9.
|
Explore Courses for Class 9 exam
|
|
Similar Class 9 Doubts
Top Courses for Class 9View all
The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr?
Question Description
The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr?.
The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr?.
Solutions for The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr? in English & in Hindi are available as part of our courses for Class 9.
Download more important topics, notes, lectures and mock test series for Class 9 Exam by signing up for free.
Here you can find the meaning of The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr? defined & explained in the simplest way possible. Besides giving the explanation of
The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr?, a detailed solution for The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr? has been provided alongside types of The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr? theory, EduRev gives you an
ample number of questions to practice The bisectors of base angles q and R of an equilateral triangle pqr intersect at point S. ST and smdrawn parallel to the sides PQ and PR intersecting side q r at t and M respectively prove that QT = cm equals Mr? tests, examples and also practice Class 9 tests.
|
|
Explore Courses for Class 9 exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup