Answer:
Tile 1:
In any triangle (regardless its type), the sum of measures of the internal angles is 180°.
This means that:
∠ABC + ∠BAC + ∠ACB = 180°
Tile 2:
The sum of measures of internal angles of a triangle is 180°.
We are given that:
ΔABC is isosceles where AB = AC
This means that:
∠ABC = ∠ACB
We are also given that measure angle BAC is 70 degrees
180 = ∠ABC + ∠ACB + 70
∠ABC + ∠ACB = 110°
We know that both angles are equal, therefore:
∠ABC = ∠ACB = 110/2 = 55°
Tile 3:
We are given that ΔQPR is an isosceles triangle where PQ = QR
This means that:∠QPR = ∠QRP
We are given that ∠QRP = 30°
This means that:∠QPR = 30°
Tile 4:
A diagram representing the given scenario is attached.
Now we have:
point D is midpoint to AB and point E is midpoint to BC
There is a theorem stating that: "In a triangle, a line joining the midpoints of two sides is parallel to the third side and equals half its length"
Applying this to the givens, we would conclude that:ED is parallel to AC
Now, since these two lines are parallel, then angles BAC and BDE are corresponding angles which means that they are equal.
This means that:∠BAC = ∠BDE = 45°