Given: Triangles ABC and DBC are isosceles, m∠BDC = 30°, and m∠ABD = 155°. Find m∠ABC, m∠BAC, and m∠DBC.
m∠DBC = (180-30)/2 = 75°
m∠ABC = m∠ABD - m∠DBC = 155-75 = 80°
so m∠A = (180 - 2*80) = 20°
Now you know all the angles.
To find the measures of the angles in the given triangles, let's use the properties of isosceles triangles and the fact that the sum of the angles in a triangle is 180 degrees.
Let's start by finding the measure of angle BDC. We are given that m∠BDC = 30°.
Since triangle DBC is isosceles, we know that angles BDC and BCD are congruent. Since the sum of the angles in triangle DBC is 180 degrees, we can find the measure of angle BCD:
m∠BCD = 180° - m∠BDC - m∠DBC
m∠BCD = 180° - 30° - m∠DBC
m∠BCD = 150° - m∠DBC
Since triangle ABC is isosceles, we know that angles ABC and ACB are congruent. Let's find the measure of angle BCD in terms of m∠ABC:
m∠BCD = m∠ABC
Now, we can set up an equation using the fact that the sum of the angles in triangle ABC is 180 degrees. We know that:
m∠ABC + m∠ABC + m∠BAC = 180°
Simplifying this equation, we get:
2m∠ABC + m∠BAC = 180°
Now, substitute m∠BCD = m∠ABC into the equation:
2m∠BCD + m∠BAC = 180°
2(150° - m∠DBC) + m∠BAC = 180°
300° - 2m∠DBC + m∠BAC = 180°
-2m∠DBC + m∠BAC = -120°
Now, we have two equations:
2m∠ABC + m∠BAC = 180°
-2m∠DBC + m∠BAC = -120°
Solve these equations simultaneously to find the values of the angles m∠ABC, m∠BAC, and m∠DBC.