Question

find x and y.
mild
isosceles triangles
find the measure
of ∠c if ab = bc.
medium
find x and y.
medium

Explanation:

Top Triangle (Mild)

Step1: Use isosceles triangle property

In an isosceles triangle, base - angles are equal. So, \(y = 62\).

Step2: Use angle - sum property of a triangle

The sum of angles in a triangle is \(180^{\circ}\). So, \(62+62 + 2(x + 1)=180\).
Simplify the left - hand side: \(124+2x + 2=180\), which becomes \(2x+126 = 180\).
Subtract 126 from both sides: \(2x=180 - 126=54\).
Divide by 2: \(x = 27\).

Middle Triangle (Medium)

Step1: Use isosceles triangle property

Since \(AB = BC\), then \(\angle A=\angle C\). So, \(3x + 7=2x+6\).
Subtract \(2x\) from both sides: \(3x-2x+7=2x - 2x+6\), which gives \(x+7 = 6\).
Subtract 7 from both sides: \(x=-1\).

Step2: Find \(\angle C\)

Substitute \(x = - 1\) into \(\angle C=2x + 6\). Then \(\angle C=2(-1)+6=4\).

Bottom Triangle (Medium)

Step1: Use isosceles triangle property

In an isosceles right - triangle (right - angled and two sides equal), the two non - right angles are equal. So, \(2x-11=y + 16\).
Also, using the angle - sum property of a triangle (\(90+(2x-11)+(y + 16)=180\)). Simplify the angle - sum equation: \(90+2x-11+y + 16=180\), which is \(2x+y+95 = 180\), or \(2x+y=85\).
From \(2x-11=y + 16\), we can express \(2x=y + 27\).
Substitute \(2x=y + 27\) into \(2x+y=85\): \((y + 27)+y=85\).
Simplify: \(2y+27 = 85\). Subtract 27: \(2y=58\), so \(y = 29\).
Substitute \(y = 29\) into \(2x=y + 27\). Then \(2x=29+27 = 56\), so \(x = 28\).

Answer:

• Top Triangle: \(x = 27\), \(y = 62\)
• Middle Triangle: \(\angle C=4^{\circ}\)
• Bottom Triangle: \(x = 28\), \(y = 29\)