1995 AMC 8 Problem 13

Attempt Problem 13 of the 1995 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1995 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

13.

In the figure,

\angle A,

\angle B

and

\angle C

are right angles. If

\angle AEB = 40^\circ

and

\angle BED = \angle BDE,

then

\angle CDE =

75^\circ

80^\circ

85^\circ

90^\circ

95^\circ

Answer: E

Concepts:angle chasing angle sum

Difficulty rating: 1150

Small hint:

In triangle

BDE,

the angles at

E

and

D

are equal and

\angle B = 90^\circ

Big hint:

\angle AED = \angle AEB + \angle BED;

then use quadrilateral

AEDC,

whose angles sum to

360^\circ

Solution:

In triangle

BDE,

the angles at

E

and

D

are equal and

\angle B = 90^\circ,

so

\angle BED = \angle BDE = 45^\circ.

Then

\angle AED = \angle AEB + \angle BED

= 40^\circ + 45^\circ

= 85^\circ.

In quadrilateral

AEDC,

the angles at

A

and

C

are

90^\circ,

so

\begin{aligned} \angle CDE &= 360^\circ - 90^\circ - 90^\circ \\ &\quad {}- 85^\circ \\ &= 95^\circ. \end{aligned}

Thus, the correct answer is E .