what are the values of t and u? (triangle with vertices a, b, c, angle at c is 51°, sides ac and ab marked equal, angle at a is t, angle at b is u)

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# Explanation:
## Step1: Identify the triangle type
The triangle has two equal sides (marked with red ticks), so it's isosceles. In an isosceles triangle, the angles opposite equal sides are equal. Angle at $ C $ is $ 51^\circ $, and angle at $ A $ ($ t $) is opposite the side equal to the side opposite angle $ C $? Wait, no—wait, the equal sides are $ AC $ and $ AB $? Wait, no, the marks: the side $ AC $ and $ AB $? Wait, no, looking at the triangle: vertices $ A $, $ B $, $ C $. The marks are on $ AC $ and $ AB $? Wait, no, the two sides with ticks: one is on $ AC $, the other on $ AB $? Wait, no, maybe $ AC $ and $ BC $? Wait, no, the triangle: side $ AC $ and side $ AB $ have ticks? Wait, no, the diagram: from $ A $, one side to $ C $ (with a tick), and from $ A $ to $ B $ (with a tick)? Wait, no, the two sides with ticks are $ AC $ and $ AB $? Wait, no, the angle at $ C $ is $ 51^\circ $, and the sides opposite angles: in triangle $ ABC $, if $ AC = AB $, then angles opposite them (angle $ B $ and angle $ C $) would be equal? Wait, no, let's correct: in triangle $ ABC $, if sides $ AC $ and $ AB $ are equal (ticks), then the angles opposite them are angle $ B $ (opposite $ AC $) and angle $ C $ (opposite $ AB $)? Wait, no, side $ AC $ is opposite angle $ B $, side $ AB $ is opposite angle $ C $. So if $ AC = AB $, then angle $ B = $ angle $ C $. Wait, angle $ C $ is $ 51^\circ $, so angle $ B $ (which is $ u $) would be $ 51^\circ $? Wait, no, maybe I got the sides wrong. Wait, the two sides with ticks: one is on $ AC $, the other on $ BC $? Wait, no, the diagram: vertex $ A $, $ B $, $ C $. The side from $ A $ to $ C $ has a tick, and the side from $ B $ to $ A $ has a tick? Wait, no, the two sides with ticks are $ AC $ and $ AB $, so triangle $ ABC $ is isosceles with $ AC = AB $. Therefore, angles opposite these sides: angle $ B $ (opposite $ AC $) and angle $ C $ (opposite $ AB $) are equal. Wait, angle $ C $ is $ 51^\circ $, so angle $ B $ ( $ u $) is $ 51^\circ $? Then angle $ A $ ( $ t $) is $ 180 - 51 - 51 = 78^\circ $? Wait, no, maybe the equal sides are $ AC $ and $ BC $. Wait, let's re-express: in triangle $ ABC $, the two sides with ticks are $ AC $ and $ BC $, so $ AC = BC $. Then angles opposite them: angle $ B $ (opposite $ AC $) and angle $ A $ (opposite $ BC $) are equal. So angle $ A = $ angle $ B $. Wait, angle $ C $ is $ 51^\circ $, so sum of angles is $ 180 $. So angle $ A + $ angle $ B + 51 = 180 $, and angle $ A = $ angle $ B $, so $ 2t + 51 = 180 $, so $ 2t = 129 $, $ t = 64.5 $? No, that can't be. Wait, maybe I misidentified the equal sides. Let's look again: the diagram shows two sides with ticks: one on $ AC $ (from $ A $ to $ C $) and one on $ AB $ (from $ A $ to $ B $)? Wait, no, the two sides with ticks are $ AC $ and $ AB $, so $ AC = AB $. Then angle at $ C $ ( $ 51^\circ $) is opposite $ AB $, and angle at $ B $ ( $ u $) is opposite $ AC $. Since $ AC = AB $, angle $ B = $ angle $ C $, so $ u = 51^\circ $. Then angle at $ A $ ( $ t $) is $ 180 - 51 - 51 = 78^\circ $. Wait, that makes sense. So:

## Step1: Determine equal angles (isosceles triangle)
Since $ AC = AB $ (ticks), $ \angle B = \angle C $. $ \angle C = 51^\circ $, so $ u = 51^\circ $.

## Step2: Calculate $ t $ (angle at $ A $)
Sum of angles in a triangle: $ t + u + 51 = 180 $. Substitute $ u = 51 $: $ t + 51 + 51 = 180 $ → $ t + 102 = 180 $ → $ t = 180 - 102 = 78 $.

# Answer:
$ t = \boxed{78}^\circ $
$ u = \boxed{51}^\circ $

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the triangle type

Step1: Determine equal angles (isosceles triangle)

Step2: Calculate \( t \) (angle at \( A \))

Answer: