Sensim Math · Depth 한국어

4-2 · Triangles

Chain isosceles base angles to find unknown angles

4.MD.C.74.G.A.2 · take · grade 4

Archetype: Isosceles and Equilateral Angle Chaining · step in a 6-type progression

▶ Practice — 10 problems

In the figure, sides ABAB, ACAC, and CDCD have equal length. Find the measure of angle BACBAC.

A B C D 35°
Show solution

Understand

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.

Givens
  • AB = AC = CD (three equal segments)
  • B, C, D lie on the base in that order, so B, C, D are collinear
  • Angle ADC = 35 degrees
Unknowns
  • The measure of angle BAC
Constraints
  • An isosceles triangle has two equal base angles
  • Angles in a triangle sum to 180 degrees
  • Angles on a straight line sum to 180 degrees

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram#11 Work Backwards

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.

Execute

#7 Identify Subproblems 4.MD.C.7
Since AC = CD, triangle ACD is isosceles with equal base angles at A and D, so angle DAC = angle ADC = 35 degrees. Then angle ACD = 180 - 35 - 35 = 110 degrees.
∠ACD=180∘−35∘−35∘=110∘\angle ACD = 180^\circ - 35^\circ - 35^\circ = 110^\circ
Equal sides give equal base angles, then the triangle-sum gives the third angle.
#7 Identify Subproblems 4.MD.C.7
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∘\angle ACB = 180^\circ - 110^\circ = 70^\circ
Two angles that sit on a straight line always add to 180 degrees.
#7 Identify Subproblems 4.MD.C.7
Since AB = AC, triangle ABC is isosceles with equal base angles at B and C, so angle ABC = angle ACB = 70 degrees. Then angle BAC = 180 - 70 - 70 = 40 degrees.
∠BAC=180∘−70∘−70∘=40∘\angle BAC = 180^\circ - 70^\circ - 70^\circ = 40^\circ
The same equal-base-angle rule, applied to the second isosceles triangle, finishes the chain.
Answer: 40 degrees

Review

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.

Draw the figure to scale (tool 1) and measure angle BAC with a protractor to confirm the 40-degree answer found by stepwise angle chasing.

Standards · min grade 4

  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Chaining isosceles base angles, the triangle-sum, and the straight-line angle across two triangles to find angle BAC.
💡 This only needs Grade 4 angle-adding and the equal-base-angle rule you already know!