The measure of angle ABD if triangles ACD and BCD are isosceles is 138 degrees
Isosceles triangles are triangles that have their base angles to be equal. From the given diagram;
m<BAC = 18 degrees
m<BDC = 48 degrees
Required angle
m<ABD
From the diagram, m<DAB = m<BAC = 18 degrees
Also, m<ADC + m<ACD + m<DAC = 180
Since triangles ACD is isosceles, hence m<ADC = m<ACD. The expression above becomes
m<ADC + m<ADC + m<DAC = 180
2m<ADC + m<DAC = 180
2m<ADC = 180 - 36
2m<ADC = 144
m<ADC = 144/2
m<ADC = 72 degrees
This means that m<ADB = 72 - m<BDC
m<ADB = 72 - 48
m<ADB = 24 degrees
Get the required angle ABD
From triangle ADB, m<ABD + m<ADB + m<DAB = 180
m<ABD + 24 + 18 = 180
m<ABD + 42 = 180
m<ABD = 180 - 42
m<ABD = 138 degrees
Hence the measure of angle ABD if triangles ACD and BCD are isosceles is 138 degrees.
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