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°