SOLUTION: Suppose that M lies on the side BC of triangle ABC, and angle BAM = angle CAM = a. Let the line through C parallel to MA meet BA produced at P. Prove that triangle APC is isosce

Question 1113548: Suppose that M lies on the side BC of triangle ABC, and angle BAM = angle CAM = a. Let the line through C parallel to MA meet BA produced at P.
Prove that triangle APC is isosceles and hence AP = AC
Found 2 solutions by mananth, greenestamps:
Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!

.
MA is parallel to PC
and
AC is the transversal
Therefore CAM = angle ACP ( alternate angles)
MA is parallel to PC
and
BP is the transversal
BAM =APC ( corresponding angles)
But BAM = CAM =a
therefore APC = ACP
Therefore CAP is an isosceles triangle
hence AP = AC

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

A different approach than used by the other tutor....

(1) angle MAC is congruent to angle ACP -- alternate interior angles
(2) angle BAM is congruent to angle CAM -- given
(3) angle PAC plus angle BAC = 180 degrees -- straight angle
(4) angle PAC plus angle ACP plus angle CPA = 180 degrees -- sum of angles of a triangle
(5) angle APC is congruent to angle ACP -- (3) and (4)
(6) triangle APC is isosceles -- two congruent angles
(7) AC = AP -- sides opposite congruent angles in an isosceles triangle
RELATED QUESTIONS
ABC is a right angle isosceles triangle, angle BCA = 90, with BC as the base and AB as... (answered by ikleyn)
ABC is a right angle isosceles triangle, angle BCA = 90, with BC as the base and AB as... (answered by Boreal,greenestamps)
A triangular lot ABC with BC=400m and B=50° is divided into two parts by the line... (answered by CPhill)
in right triangle abc, side bc = 9 and side ac = 7.find the m angle a to the nearest... (answered by Alan3354)
Could someone outline a strategy for solving the following problem? *Let \( \triangle... (answered by CPhill)
In triangle ABC, M is the midpoint of line AB. Let D be the point on line BC such that... (answered by math_helper)
In right triangle ABC, AB= 20, AC = 12, BC = 16, and m angle C = 90. Find, to the nearest (answered by Fombitz)
In tetrahedron ABCO,angle AOB = angle AOC = angle BOC = 90degree.A cube is inscribed in... (answered by Alan3354)
Sir, This is quenstion on congruence of Triangles.ABC is a triangle in which AD is the... (answered by longjonsilver)