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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?
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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.
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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?
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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?.
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