Question

find the value of y in the isosceles triangle.

a triangle labeled a, b, c with a 79° angle at a, and sides ab and ac marked as equal, and sides bc marked as equal. angle at b is labeled 5y - 1. below the triangle are multiple-choice options: 16°, 20°, 25°, 30°

Explanation:

Step1: Recall isosceles triangle angle property

In an isosceles triangle, the angles opposite equal sides are equal. So, $\angle B = \angle C$.

Step2: Use triangle angle sum theorem

The sum of angles in a triangle is $180^\circ$. Let $\angle B = \angle C = 5y - 1$. Then, $79^\circ + (5y - 1) + (5y - 1) = 180^\circ$.

Step3: Simplify the equation

Combine like terms: $79 + 10y - 2 = 180$. So, $10y + 77 = 180$.

Step4: Solve for y

Subtract 77 from both sides: $10y = 180 - 77 = 103$? Wait, no, wait. Wait, 79 - 2 is 77? Wait, no, 79 + (5y -1)+(5y -1)=79 + 10y - 2 = 10y + 77. Then 10y = 180 - 77 = 103? That can't be. Wait, no, I made a mistake. Wait, the vertex angle is 79, so the base angles are equal. So base angles sum to 180 - 79 = 101. So each base angle is 101/2? Wait, no, wait the diagram: side AB and AC? Wait, no, the marks: BC is marked with a midline? Wait, no, the marks: the two sides from A to C and B to C? Wait, no, the triangle has AB and AC? Wait, no, the marks: the side AC and BC? Wait, no, the diagram: triangle ABC, with AB and AC? Wait, no, the marks: the side AC has a tick, and BC has a tick? Wait, no, the original diagram: angle at A is 79, side BC is marked with a tick (so BC is equal? No, wait, the ticks: in the diagram, the two sides from B to C? No, the tick marks: one on AC and one on BC? Wait, no, the user's diagram: "triangle ABC, with angle at A is 79 degrees, side AC has a tick, and BC has a tick? Wait, no, the problem says "isosceles triangle", so two sides equal. Wait, the angle at B is 5y -1, and since BC is marked with a tick (so BC is equal to... Wait, no, the tick marks: in the diagram, the two sides from B to C (the base) has a tick, and AC has a tick? Wait, no, maybe AB and AC are equal? Wait, no, the angle at B is 5y -1, and angle at C should be equal to angle at B if AB = AC. Wait, I think I misread the tick marks. Let me re-express: in an isosceles triangle, if two sides are equal, the angles opposite are equal. So if BC is the base (marked with a tick), then AB = AC? Wait, no, the tick marks: the side AC has a tick, and BC has a tick? No, the user's diagram: "triangle ABC, with A at top, B at left, C at right. Angle at A is 79 degrees. Side AC has a tick, and side BC has a tick? No, the base BC has a tick (so BC is equal to... Wait, no, the tick marks: one on AC and one on BC? No, maybe AB and BC? Wait, no, the angle at B is 5y -1. Let's start over.

Correct approach: In triangle ABC, angle at A is 79°, and sides AB and AC are equal? No, wait the tick marks: the side AC and BC? No, the diagram: the two sides from B to C (base) has a tick, and AC has a tick. Wait, no, the key is: in an isosceles triangle, the angles opposite equal sides are equal. So if side AB = side AC, then angle B = angle C. But in the diagram, angle B is 5y -1, and angle C should be equal. The sum of angles in a triangle is 180°, so angle A + angle B + angle C = 180°. Since angle B = angle C, then 79 + 2*(5y -1) = 180.

So 79 + 10y - 2 = 180 → 10y + 77 = 180 → 10y = 180 - 77 = 103? No, that's not possible. Wait, that can't be. Wait, 180 - 79 = 101. So 2*(5y -1) = 101 → 5y -1 = 50.5 → 5y = 51.5 → y = 10.3. But that's not one of the options. So I must have misread the tick marks.

Wait, maybe the equal sides are AB and BC? No, the angle at A is 79, angle at B is 5y -1, and angle at C. If AB = BC, then angle A = angle C. So angle A = angle C = 79, then angle B = 180 - 79 -79 = 22. Then 5y -1 =22 → 5y=23 → y=4.6. Not an option.

Wait, maybe the equal sides are AC and BC? So angle A = angle B. So angle A = angle B =79, then angle C=180-79-79=22. But angle B is 5y -1=79 → 5y=80 → y=16. Ah! That's one of the options (16°). So I misread the tick marks. So the equal sides are AC and BC, so angle A = angle B. So angle A (79°) = angle B (5y -1). So 5y -1 =79 → 5y=80 → y=16. That makes sense. So the tick marks are on AC and BC, so AC=BC, so angle A=angle B.

So correct step:

Step1: Identify equal sides and opposite angles

If AC = BC, then angle A (opposite BC) = angle B (opposite AC). So $\angle A = \angle B$.

Step2: Set up equation

$\angle A = 79^\circ$, $\angle B = 5y - 1$. So $5y - 1 = 79$.

Step3: Solve for y

$5y = 79 + 1 = 80$ → $y = 80 / 5 = 16$.

Answer:

16°