Triangles — Telangana (SSC) Class 9 Mathematics Solutions (Free)
Free step-by-step Telangana (SSC) Class 9 Mathematics solutions for "Triangles" — important questions with detailed answers, download PDF for board exam preparation.
TL;DR: Free step-by-step Telangana (SSC) Class 9 Mathematics solutions for "Triangles" — important questions with detailed answers, download PDF for board ex…
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Q1: State the SSS (Side-Side-Side) congruence criterion and apply it: If in triangle ABC, AB = 3 cm, BC = 4 cm, CA = 5 cm and in triangle PQR, PQ = 3 cm, QR = 4 cm, RP = 5 cm, prove that triangle ABC ≅ triangle PQR.
Step 1: SSS congruence criterion states that if all three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent.
Step 2: In triangle ABC:
AB = 3 cm
BC = 4 cm
CA = 5 cm
Step 3: In triangle PQR:
PQ = 3 cm
QR = 4 cm
RP = 5 cm
Step 4: Comparing corresponding sides:
AB = PQ = 3 cm
BC = QR = 4 cm
CA = RP = 5 cm
Step 5: Since all three sides of triangle ABC are equal to corresponding sides of triangle PQR, by SSS criterion, triang…
Q2: Apply the SAS (Side-Angle-Side) congruence criterion: In triangles ABD and CBD, AB = CB, angle ABD = angle CBD, and BD is common. Prove that triangle ABD ≅ triangle CBD.
Step 1: SAS congruence criterion states that if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.
Step 2: In triangle ABD:
AB = given
BD = common side
angle ABD = given
Step 3: In triangle CBD:
CB = given
BD = common side
angle CBD = given
Step 4: Comparing the triangles:
AB = CB (given)
angle ABD = angle CBD (given)
BD = BD (common side)
Step 5: The included angle between si…
Q3: Apply the ASA (Angle-Side-Angle) congruence criterion: In triangles PQR and XYZ, angle Q = angle Y, QR = YZ, and angle R = angle Z. Prove that triangle PQR ≅ triangle XYZ.
Step 1: ASA congruence criterion states that if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of another triangle, then the triangles are congruent.
Step 2: In triangle PQR:
angle Q = given
QR = given (included side between angles Q and R)
angle R = given
Step 3: In triangle XYZ:
angle Y = given
YZ = given (included side between angles Y and Z)
angle Z = given
Step 4: Comparing the triangles:
angle Q = angle Y (given)
QR = YZ …
Q4: Apply the RHS (Right angle-Hypotenuse-Side) congruence criterion: In right triangle ABC with right angle at B and right triangle DEF with right angle at E, if AC = DF (hypotenuses) and AB = DE (one side), prove that triangle ABC ≅ triangle DEF.
Step 1: RHS congruence criterion states that if the hypotenuse and one side of a right triangle are equal to the hypotenuse and corresponding side of another right triangle, then the triangles are congruent.
Step 2: In triangle ABC:
Angle B = 90° (given)
AC = hypotenuse
AB = one side
Step 3: In triangle DEF:
Angle E = 90° (given)
DF = hypotenuse
DE = one side
Step 4: Comparing the triangles:
Angle B = Angle E = 90° (both right angles)
AC = DF (given hypotenuses are equal)
AB = DE (given, corr…
Q5: In triangle ABC, if AB = AC, prove that the angles opposite to these sides are equal, i.e., angle C = angle B.
Step 1: Given: AB = AC in triangle ABC. To prove: angle C = angle B (angles opposite equal sides are equal).
Step 2: Draw the angle bisector of angle A, and let it meet BC at point D.
Step 3: In triangle ABD and triangle ACD:
AB = AC (given)
AD = AD (common side)
angle BAD = angle CAD (AD is angle bisector of angle A)
Step 4: By SAS congruence criterion, triangle ABD ≅ triangle ACD.
Step 5: From the congruence, corresponding angles are equal:
angle ABD = angle ACD
angle ADB = angle ADC
BD = …
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