Need help understanding triangle properties problem

Question

I am struggling with a problem that involves triangle properties, and I am hoping someone can help me understand it better. Here is the problem:

Two altitudes of a triangle do not intersect, and the acute angle between their extensions is $45^{\circ}$. Then,

(a) one of the angles of the triangle is $45^{\circ}$;

(b) one of the angles of the triangle is $135^{\circ}$;

(c) it is impossible to determine;

(d) there is no such triangle.

I have attempted to solve the problem by drawing a triangle and its two altitudes, but I'm not sure where to go from there. I think I need to use the fact that the product of the lengths of the two altitudes is equal to the product of the sides of the triangle, but I'm not sure how that helps me determine the angles.

Can someone please help me understand this problem better? Thank you in advance!

Answer 1

When two altitudes of a triangle do not intersect, it implies that the triangle is obtuse. In this case, where one of the angles is 45°, the other angle is also 45° which is the supplementary angle of the original triangle. This ultimately means that one of the angles of the original triangle is 135°.

Answer 2

The answer is (c) it is impossible to determine.

We can however determine that one of the angles in the triangle will be either $135^o$ or $45^o$. This is because there are two types of triangle that meet the criteria, as shown in the diagram below.

Answer 3

The answer depends on which altitudes are extended.

In the diagram at left, $\overline{AE}$ is the altitude to the extension of side $\overline{BC}$ of $\triangle ABC$ and $\overline{BD}$ is the altitude to the extension of side $\overline{AC}$ of $\triangle ABC$. The extensions of these altitudes meet at a $45^\circ$ angle (I used the points $A(-2, 2)$, $B(4, 0)$, $C(0, 0)$, $D(2, -2)$, $E(-2, 0)$, and $F(-2, 6)$, so you can verify this claim). Since the acute angles of a right triangle are complementary, the measure of angle $FAD$ is $45^\circ$. For the same reason, the measure of angle $ACE$ is also $45^\circ$. Since $\angle ACB$ is supplementary to angle $ACE$, its measure is $135^\circ$. Notice that this precludes any other angle in $\triangle ABC$ from having an angle measure of $45^\circ$ since the angle sum of the interior angles of a triangle cannot exceed $180^\circ$.

In the diagram at right, $\overline{AE}$ is the altitude to the extension of side $\overline{BC}$ of $\triangle ABC$ and $\overline{CD}$ is the altitude to side $\overline{AB}$ of the same triangle. The extensions of these altitudes meet at an angle of $45^\circ$ (I used the points $A(-2, 4)$, $B(2, 0)$, $C(0, 0)$, $D(1, 1)$, $E(-2, 0)$, and $F(-2, -2)$, so you can verify this). Since the acute angles of a right riangle are complementary, the measure of angle $ECF$ is $45^\circ$. Since vertical angles are congruent, the measure of angle $DCB$ is $45^\circ$. Since the acute angles of a right triangle are complementary, the measure of angle $ABC$ is $45^\circ$. Notice that this precludes any other angle in $\triangle ABC$ from having an angle measure of $135^\circ$ since the angle sum of the interior angles of a triangle cannot exceed $180^\circ$.

Therefore, the answer cannot be determined from the given information.

Answer 4

In the figure on the left, given $\triangle ABC$ with altitudes $AD$, $CF$ meeting at $E$, and $\angle AEC=45^o$, then since $EFBD$ is a cyclic quadrilateral$$\angle ABC=\angle FBD=180^o-45^o=135^o$$And since $\angle CAB+\angle BCA=45^o$, then each is less than $45^o$.

But in the figure on the right, if altitudes $D'B'$, $C'F'$ meet at $E'$ and $\angle D'E'C'=45^o$, then in $\triangle A'B'C'$ $\angle C'A'B'=45^o$. But since exterior $\angle B'A'G=135^o$, then opposite interior obtuse $$\angle A'B'C'<135^o$$

Therefore (c), "it is impossible to determine," is true.

Answer 5

Note quadrilateral $FDCE$ is cyclic because $\angle FDC + \angle FEC = 90^\circ + 90 ^\circ = 180 ^\circ$. That means $\angle ECD = \angle ACB = 135 ^\circ$. Therefore the answer is (b) one of the angles of the triangle is $135 ^\circ$.