Triangles(9th Grade > Mathematics ) Questions and answers for exam Preparation

9th Grade > Mathematics

TRIANGLES MCQs

Total Questions : 56 | Page 6 of 6 pages

If in the given figure, ABCD is a rectangle and DCE is an equilateral triangle, then which of the following statements is/are correct?

$Δ D A E ≅ Δ C B E$
$∠ D A E = 15^{\circ}$
$Δ A E B$ is isosceles.
$Δ D A E$ is isosceles.

Discuss Question

Answer: Option A. ->

Δ D A E ≅ Δ C B E

:
A and C

∠ E D C

∠ E C D

(Angles of an equilateral triangle)
And

∠ A D C

∠ B C D

(Angles of a rectangle)

∴ ∠ E D C + ∠ A D C = ∠ E C D + ∠ B C D

\Rightarrow ∠ E D A = ∠ E C B

Δ D A E

and

Δ C B E

D E = C E

(Sides of an equilateral triangle)

D A = B C

(Sides of a rectangle)

∠ E D A

∠ E C B

(proved)
Hence,

Δ D A E ≅ Δ C B E

(SAS congruence)
Also, EA = EB (C.P.C.T.C.)

\Rightarrow Δ E A B

is an isosceles triangle.
It is given that, ABCD is a rectangle.

\Rightarrow D A \neq D C

Also, DEC is an equilateral triangle.

\Rightarrow D C = D E

∴ D A \neq D E

Hence, DAE is not an isosceles triangle.
Also, it can not be proved that

∠ D A E = 15^{\circ}

Question 52.

In the figure, AB = AC and AP $⊥$ BC. Then

AB = AP
AB < AP
AB > AP
AB < BP

Discuss Question

Answer: Option C. -> AB > AP
:
C

$A P ⊥ B C$
$⟹ △ A P B$ is a right-angled triangle.
$⟹$ AB is the hypotenuse and hence the longest side, which makes is greater than AP.
$∴ A B > A P$

Question 53.

In the given figure, AC = CE and AB $∥$ ED. The value of $x$ is ___ units.

Discuss Question

Answer: Option C. -> AB > AP
:

In Δ A B C and Δ E D C, A C = C E (given) ∠ B A C = ∠ D E C (since AB||DE and AE is a transversal, so they are alternate angles) ∠ A C B = ∠ E C D (vertically opposite angles) ∴ Δ A B C ≅ Δ E D C (A.S.A. congruence criteria) ∴ A B = D E (sides of congruent triangles) ∴ x + 10 = 2 x - 5 \Rightarrow x = 15 u n i t s

Question 54.

Which of the following options is/are appropriate criteria for congruence?

S.S.S.
A.A.A.
A.S.A.
S.A.S.

Discuss Question

Answer: Option C. -> A.S.A.
:
A, C, and D
Angle-Angle-Angle (A.A.A.) is not a criterion for congruence.
Knowing the measures of all the angles of two triangles is not sufficient to determine if the two triangles are congruent. Because two triangles having the same set of angles can be of different sizes, as shown below.

Which Of The Following Options Is/are Appropriate Criteria F...

The remaining options - Side-Side-Side (SSS), Angle-Side-Angle (ASA) and Side-Angle-Side (SAS) are criteria to determine if the given triangles are congruent.

Question 55.

$If Δ A B C ≅ Δ P Q R and it is given that ∠ A = 60^{\circ}, ∠ B = 70^{\circ} . Then the value of ∠ R is$ ___ $^{\circ}$ .