Question

*****50 POINTSSSS*****

In the figure, quadrilateral GERA is inscribed in circle P. Line TA is tangent to circle P at A, m∠REG = 78°, the measure of arc AR = 46°, and segment ER ≅ segment GA. Find each of the following measures.


m∠GAR =


m∠TAR =

measure of arc GE =

measure of arc AG =

m∠GAN =

measure of arc RE =

Answer 1

Given

• m∠REG = 78°
• mAR = 46°
• ER ≅ GA

Solution

• m∠GAR = 180° - m∠REG = 180° - 78° = 102° (supplementary angles sum to 180°)
• m∠TAR = 1/2mAR = 1/2(46°) = 23° (tangent chord angle is half the size of intercepted arc)
• m∠GAN = 180° - (m∠TAR + m∠GAR) = 180° - (23° + 102°) = 55° (straight angle is 180°)
• mAG = 2m∠GAN = 2(55°) = 110°
• mRE = mAG = 110° (as ER ≅ GA)
• mGE = 360° - (mAG + mAR + mRE) = 360° - (110° + 46° + 110°) = 94° (full circle is 360°)