Question

∆XYZ is an isosceles triangle with XZ=YZ and ZT is the angle bisector of angle XZY
.Show that ∆XZT= ∆YZT
.What would be the measures of angle XTZ and angle YTZ? ​

Answer 1

In ∆XYZ, since XZ=YZ and ZT is the angle bisector of angle XZY, by the Angle Bisector Theorem, we have XZ/XT = YZ/YT. Since XZ=YZ, this means that XT = YT.

Since we have that XZ = YZ and XT = YT, by SAS congruence we can say ∆XZT= ∆YZT.

• The measures of angle XTZ and angle YTZ in ∆XZT and ∆YZT respectively would be equal to 90 degrees because ZT is the angle bisector of angle XZY and it bisects it into two congruent angles.

• The Angle Bisector Theorem states that in any triangle, if a line is drawn from a vertex of the triangle to the midpoint of the opposite side, then the ratio of the length of this line segment to the length of the side it bisects is equal to the ratio of the length of the other two sides of the triangle.

• Formally, if in a triangle ABC, D is a point on side BC such that AD is the angle bisector of angle A, then we have: BD/DC = AD/AB

• This theorem can be proven by using similar triangles. Since AD is the angle bisector of angle A, it divides angle A into two congruent angles. Therefore, triangles ABD and ADC are similar. Therefore, their corresponding sides are in proportion.

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