Hello, Rinnie!
The last one is true.
First, we write the convese and see if it is also true,
. . or try to find a counterexample.
If the quadrilateral is an isosceles trapezoid,
. . then a pair of base angles are congruent. Converse:
If a quadrilateral has a pair of equal base angles, it is an isosceles trapezoid.
Code:
*
* *
* *
* *
* *
* *
* *
* θ θ *
* * * * * * * *
This is a quarilateral with equal base angles,
. . but it is not an isosceles trapezoid.
The converse is false.
If the quadrilateral is a kite,
. . then the nonvertex angles are congruent. Converse
If the nonvertex angles of a quadrilateral are congruent,
. . the quadrilateral is a kite. Code:
*
* *
* *
* *
* θ *
* *
* θ *
* *
* *
* *
*
This is a quadrilateral with congruent nonvertex angles,
. . but it is not a kite.
The converse is false.
If the quadrilateral is a kite,
. . then the major diagonal bisects the vertex angles. Converse:
If the major diagonal of a quadrilateral bisects the vertex angles,
. . then the quadrilateral is a kite. Code:
A
*
*|*
* | *
*α | α*
* | *
* | *
B * | * D
* β | β *
* | *
*
C
There is no way to avoid drawing a kite.
We have: .
Therefore, is a kite.
The converse is true.