Problem
Chain isosceles base angles to find unknown angles
Vertex A is at the top with B, C, D along the base. Sides AB, AC, and CD are all equal, and angle ADC is 35 degrees. I must find angle BAC.
Measurement & dataGeometry
Your answer
°
Watch out In isosceles triangle ACD, it's easy to mistakenly set angle ACD, the apex where the two equal sides meet, equal to the given 35 degrees, instead of angle DAC, the other base angle.
How to solve
Strategy Identify Subproblems — The figure splits into two isosceles triangles (ACD and ABC). I solve the one I can (ACD, where the 35-degree angle lives) first, carry the result across the straight base, then finish in triangle ABC. Working from the known angle toward the target angle keeps each step grounded.
1STEP 1
Find angle ACD in isosceles triangle ACD
AC = CD makes triangle ACD isosceles, so angle DAC = angle ADC = 35°, and angle ACD = 180 - 35 - 35 = 110 degrees.
∠ ACD = 180° - 35° - 35° = 110°
Equal sides give equal base angles, then the triangle-sum gives the third angle.
4.MD.C.7Identify SubproblemsIn triangle ACD, the angle at C (angle ACD) comes out to 110 degrees.
Why?
The three corners of triangle ACD share a fixed total of 180 degrees, so the corner at C is that total with the other two corners taken away.
Why?
In any flat triangle the three corner angles always fill exactly one straight angle of 180 degrees.
🧱A triangle's angles sum to 180 degreesThe three corners of any flat triangle always add up to one straight angle, 180 degrees.
Why?
The 180 degrees is one whole built from the three corner angles, so the leftover corner is the whole with the two known corners removed.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
The two corners being removed, at A and at D, are 35 degrees each: the angle at D is given as 35, and the angle at A must match it.
Why?
Sides AC and CD have the same length, so the triangle folds along its middle line and the corner at A lands exactly on the corner at D, making their angles equal.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
2STEP 2
Find angle ACB using the straight base
B, C, D lie on one straight line, so angle ACB and angle ACD are a straight-line pair: angle ACB = 180 - 110 = 70 degrees.
∠ ACB = 180° - 110° = 70°
Two angles that sit on a straight line always add to 180 degrees.
4.MD.C.7Identify Subproblems3STEP 3
Find angle BAC in isosceles triangle ABC
AB = AC makes triangle ABC isosceles, so angle ABC = angle ACB = 70°, and angle BAC = 180 - 70 - 70 = 40 degrees.
∠ BAC = 180° - 70° - 70° = 40°
The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.
4.MD.C.7Identify SubproblemsAnswer
40 °
∠ BAC = 180° - 70° - 70° = 40°
Angle BAC = 40 degrees is acute, which fits the narrow top vertex of triangle ABC where two long equal sides meet. Each triangle's angles also total 180 degrees (35+35+110 and 70+70+40), confirming the chain is consistent.
Takeaway
This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!
- Find angle ACD in isosceles triangle ACD
- Find angle ACB using the straight base
- Find angle BAC in isosceles triangle ABC
Where next?
Level upsuggested
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested