Protruding Angle

Geometry Level 1

In the figure above, ∠ D A B = 11 1 ∘ \angle DAB =111^\circ . Find the value of x x .


The answer is 7.

M M by M M , 29, USA

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5 solutions

Terry Smith
Nov 29, 2016

∠ D A C = 11 1 ∘ − 4 x ∘ \angle DAC = 111 ^ \circ-4x ^ \circ

x ∘ + 9 0 ∘ + ( 111 − 4 x ) ∘ = 18 0 ∘ x ^ \circ+90 ^ \circ+(111-4x) ^ \circ=180 ^ \circ

The rest is left as an exercise for the student.

10 Helpful 2 Interesting 3 Brilliant 0 Confused

3rd line of the calculation is like written by typical mathematics professor at uni 😂 I still like it tho

Damian rot - 4 years, 6 months ago
M M
Nov 21, 2016

Using simultaneous equations:

Let ∠ D A C \angle DAC measure y degrees. Then ∠ B C A \angle BCA also measures y degrees (alternate interior angles).

By angle addition, y + 4 x = 111 y + 4x = 111 . Using same-side interior angles, ∠ D C B \angle DCB must measure 90 degrees, so x + y = 90 x + y = 90 as well.

Solving the second equation for y, y = 90 − x y = 90 - x . Substituting into the first equation, 90 − x + 4 x = 111 90 - x + 4x = 111 . Solving yields the answer x = 7.

Using expressions:

The measure of ∠ D A C \angle DAC can be rewritten as 111 - 4x. Then the three angles inside the triangle measure (111-4x), 90, and x. Writing an equation using the fact that the angles in a triangle sum to 180:

111 - 4x + 90 + x = 180

Again, solving yields the answer x = 7.

7 Helpful 0 Interesting 0 Brilliant 0 Confused

Challenging question

Michael Chait - 4 years, 6 months ago

A challenging question.

Michael Chait - 4 years, 6 months ago

I think the parallel lines are immaterial. Without this information we can solve the problem

Alfa Claresta - 4 years, 3 months ago

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They are! However, sometimes having extra information makes the problem harder to solve. You have to know what is important and what isn't.

M M - 4 years, 3 months ago

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I think if you change angle B given instead , it will be more sophisticated ...

Alfa Claresta - 4 years, 3 months ago

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@Alfa Claresta – I am not seeking to make extra calculations; the challenge in this problem is knowing what calculations to do.

M M - 4 years, 3 months ago
Irina Stanciu
Dec 30, 2016

∠ D A B = 4 x + y \angle DAB=4x+y and in the ADC right-angled triangle we've got x + y + 90 = 180 x+y+90=180 => 4 x + y − x − y = 111 − 90 4x+y-x-y=111-90 => 3 x = 21 3x=21 => x = 7 x=7

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Toshit Jain
Nov 26, 2016

Draw a line || to CD such that it meets BC at E. Angle DCA = Angle CAE = x°(alternate interior angles) Then Angle EAB = 4x°-x°=3x° Angle DAE = 180°-90°=90 Angle DAE + Angle EAB = 111° 90° + 3x° = 111 Therefore , x=7°

1 Helpful 0 Interesting 0 Brilliant 0 Confused

in Triangle ADC, ADC=90, DAC+ACD=90, 111-4x+x=90, 111-3x=90, 3x=21, x=7

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