Question

what is the value of t? e f t 34° g t = °

Explanation:

Step1: Identify triangle type

The triangle has two equal sides (marked with ticks), so it's isosceles. Thus, the base angles (at E and F) are equal? Wait, no—wait, the equal sides are EG and FG? Wait, no, the ticks are on EG and EF? Wait, no, looking at the triangle: sides EG and FG? Wait, no, the two tick marks are on EG and EF? Wait, no, the triangle is E-G-F, with EG and EF? Wait, no, the vertices are E, G, F. The side from E to G has a tick, and the side from F to G has a tick? Wait, no, the tick marks are on EG and FG? Wait, no, the diagram: E at top, G at bottom, F at right. So EG is left side, FG is right side, and EF is top. Wait, the tick marks are on EG and FG? Wait, no, the two tick marks: one on EG (left side) and one on FG (right side)? Wait, no, the problem: the triangle has two sides equal (marked by the red ticks), so it's an isosceles triangle with \( EG = FG \)? Wait, no, wait: the angle at G is 34 degrees. Wait, in an isosceles triangle, the angles opposite equal sides are equal. Wait, if \( EG = FG \), then the angles opposite them would be angles at F and E? Wait, no, let's label the triangle: vertices E (top), G (bottom), F (right). So sides: EG (left), GF (right), EF (top). The tick marks are on EG and GF? Wait, no, the two tick marks: one on EG (left) and one on GF (right)? So \( EG = GF \), so triangle EGF is isosceles with \( EG = GF \). Therefore, the base angles are at E and F? Wait, no, the angle at G is 34 degrees. Wait, in triangle EGF, if \( EG = GF \), then the angles opposite those sides: angle at F (opposite EG) and angle at E (opposite GF). Wait, no, side EG is opposite angle F, side GF is opposite angle E. So if \( EG = GF \), then angle F = angle E. Wait, but we need to find angle at F (t). Wait, the sum of angles in a triangle is 180 degrees. So angle at G is 34 degrees, angle at E and angle at F: if \( EG = GF \), then angle E = angle F? Wait, no, wait, maybe I got the equal sides wrong. Wait, maybe the equal sides are EG and EF? No, the tick marks are on EG and FG. Wait, maybe the triangle is isosceles with \( EG = EF \)? No, the tick marks are on EG and FG. Wait, let's re-express:

Wait, the triangle: E, G, F. Let's list the sides:

- EG: from E to G (left side)
- GF: from G to F (right side)
- EF: from E to F (top side)

The two tick marks are on EG and GF, so \( EG = GF \). Therefore, triangle EGF is isosceles with \( EG = GF \). Therefore, the base is EF, and the equal sides are EG and GF. Therefore, the base angles are at E and F? Wait, no, the base angles are the angles adjacent to the base. Wait, in an isosceles triangle, the angles opposite the equal sides are equal. So side EG is opposite angle F, side GF is opposite angle E. Therefore, if \( EG = GF \), then angle F = angle E. Wait, but we know angle at G is 34 degrees. So sum of angles: angle G + angle E + angle F = 180. If angle E = angle F, then 34 + 2t = 180? Wait, no, that would be if angle E and angle F are equal. Wait, but maybe I made a mistake. Wait, maybe the equal sides are EG and EF? No, the tick marks are on EG and FG. Wait, maybe the triangle is isosceles with \( EG = EF \), so angle at G and angle at F are equal? Wait, no, let's check again.

Wait, maybe the equal sides are EG and EF? No, the tick marks are on EG and FG. Wait, perhaps the correct approach: in a triangle, the sum of angles is 180. If two sides are equal, the triangle is isosceles, so the base angles are equal. Wait, the angle at G is 34 degrees. If the two equal sides are EG and FG, then the angles opposite them (angle F and angle E) are equal. Wait, no, side EG is opposite angle F, side FG is opposite angle E. So angle F = angle E. Then sum of angles: 34 + angle E + angle F = 180. Since angle E = angle F, let's call each x. So 34 + x + x = 180 → 34 + 2x = 180 → 2x = 146 → x = 73. Wait, but that would be angle E and angle F. But wait, maybe the equal sides are EG and EF, so angle at G and angle at F are equal. Wait, no, let's look again. Wait, maybe the triangle is isosceles with \( EG = EF \), so angle at G (34 degrees) and angle at F (t) are equal? No, that would be if \( EG = EF \), then angle at G = angle at F. But then sum would be 34 + 34 + angle E = 180 → angle E = 112. But that doesn't seem right. Wait, maybe I misidentified the equal sides. Wait, the tick marks: one on EG (left) and one on EF (top)? No, the diagram: E at top, G at bottom, F at right. So EG is left, GF is right, EF is top. The tick marks are on EG and GF, so \( EG = GF \). Therefore, triangle EGF is isosceles with \( EG = GF \), so angles opposite: angle F (opposite EG) and angle E (opposite GF). Therefore, angle F = angle E. Then sum: 34 + angle E + angle F = 180. So 34 + 2t = 180? Wait, no, t is angle F, so angle E is also t? Wait, no, angle at G is 34, angle at F is t, angle at E is also t? Then 34 + t + t = 180 → 2t = 146 → t = 73. Wait, but maybe the equal sides are EG and EF, so angle at G = angle at F. Wait, no, let's check the diagram again. The angle at G is 34 degrees, between sides EG and GF. So if EG and GF are equal, then it's an isosceles triangle with legs EG and GF, base EF. Therefore, the base angles are at E and F, which are equal. So angle E = angle F. Then sum: 34 + angle E + angle F = 180. So 34 + 2t = 180 → t = (180 - 34)/2 = 146/2 = 73. Wait, but maybe I got the equal sides wrong. Wait, maybe the equal sides are EG and EF, so angle at G = angle at F. Then angle F = 34, but that would make angle E = 180 - 34 - 34 = 112, but then the sides opposite would be GF and EG, which would not be equal. So that's not possible. Therefore, the correct approach is: since the triangle has two equal sides (EG = GF), it's isosceles, so angles opposite those sides (angle F and angle E) are equal. Sum of angles: 34 + t + t = 180 → 2t = 146 → t = 73. Wait, but wait, maybe the equal sides are EG and EF, so angle at G = angle at F. But that would mean angle F = 34, but then angle E = 180 - 34 - 34 = 112, and sides EG and EF would be equal, but the tick marks are on EG and GF, not EF. So the tick marks are on EG and GF, so EG = GF, so angle F = angle E. Therefore, t = 73. Wait, but let's confirm: sum of angles in a triangle is 180. So 34 + 73 + 73 = 180. 34 + 146 = 180. Yes, that works. So t is 73 degrees.

Step2: Calculate t

Sum of angles in a triangle: \( 180^\circ \).
Given \( \angle G = 34^\circ \), and \( EG = FG \) (isosceles triangle), so \( \angle E = \angle F = t \).
Thus:
\( 34^\circ + t + t = 180^\circ \)
\( 34^\circ + 2t = 180^\circ \)
Subtract \( 34^\circ \) from both sides:
\( 2t = 180^\circ - 34^\circ = 146^\circ \)
Divide by 2:
\( t = \frac{146^\circ}{2} = 73^\circ \)

Answer:

\( 73 \)