Answer:
1) (B)
2) (A)
3) (D)
4) (C)
Step-by-step explanation:
Given:
There are 4 statements and we need to find the answer of each statement.
Statement 1:
m ∠ABC + m ∠BAC + m ∠ACB when Δ ABC is an isosceles triangle
Sum of all interior angles of a triangle is always equal to 180°.
⇒ m ∠ABC + m ∠BAC + m ∠ACB = 180°
Therefore, the correct option is (B).
Statement 2:
m∠ABC when m∠BAC = 70° and ΔABC is an isosceles triangle
From the triangle ABC, ∠A = 70, ∠B = ∠C(Isosceles triangle property)
Let ∠B = ∠C = 'x'.
Sum of all interior angles of a triangle is always equal to 180°.
⇒ m ∠ABC + m ∠BAC + m ∠ACB = 180°
m ∠ABC = ∠B = 55° [Option (A)]
Statement 3:
m∠QPR when m∠QRP = 30° and ΔPQR is an isosceles triangle.
From the triangle QPR, ∠R = 30°, QP ≅ QR
Now, from isosceles triangle definition, the angles opposite to the congruent sides are also congruent. Therefore,
∠P = ∠R = 30°
Now, m∠QPR is same as ∠P. So, m∠QPR = 30° [Option D].
Statement 4:
m∠BDE when m∠BAC = 45° and points D and E are the midpoints of AB and BC, respectively, in ΔABC
From the figure shown below,
DE is the midsegment of sides AB and BC. Therefore, from midsegment theorem, DE || AC (third side).
Therefore, for two parallel line DE and AC cut by transversal AB,
m∠BDE = m∠BAC (Corresponding angles are congruent)
m∠BDE = 45° [Option C]
