Find Angle Measures Using Triangle Incenter Properties and Linear Equations

Question

If C is the incenter of △AMD, m∠AMC = (3x + 6)° and m∠DMC = (8x - 49)°, find each measure. a) x = ______ b) m∠DMC = ______ c) m∠MAD = ______ d) m∠ADM = ______ e) m∠ADC = ______

Accepted Answer

a) x = 11; b) m∠DMC = 39°; c) m∠MAD = 66°; d) m∠ADM = 48°; e) m∠ADC = 66°

Answer

Step 1: Recall that the incenter of a triangle is the intersection of the angle bisectors, so MC bisects ∠AMD. This means m∠AMC = m∠DMC. Step 2: Set up the equation 3x + 6 = 8x - 49. Solve for x: 6 + 49 = 8x - 3x → 55 = 5x → x = 11. Step 3: Substitute x = 11 into m∠DMC = 8x - 49: m∠DMC = 8(11) - 49 = 88 - 49 = 39°. Since m∠AMC = m∠DMC, m∠AMD = 39° + 39° = 78°.  Step 4: We know m∠MAD is twice the given 33° (because AC is the angle bisector of ∠MAD), so m∠MAD = 2×33° = 66°.  Step 5: Use the triangle angle sum theorem for △AMD: m∠MAD + m∠AMD + m∠ADM = 180°. Substitute known values: 66° + 78° + m∠ADM = 180° → m∠ADM = 180° - 144° = 48°.  Step 6: DC bisects ∠ADM, so m∠ADC = ½×m∠ADM = ½×48° = 24°? Correction: Wait, use triangle angle sum for △ADC: m∠DAC = 33°, m∠ACD = 90° + ½m∠AMD (incenter property: angle between angle bisectors is 90° + ½ the opposite angle) = 90 + 39 = 129°? No, better: In △ADC, m∠DAC=33°, m∠ACD = 180 - m∠AMC = 180 - 39 = 141? No, correct way: Since DC bisects ∠ADM, m∠ADC = ½×48=24? No, wait, in △DMC, we know m∠DMC=39°, m∠MCD=90+½×66=123? No, let's use angle sum for △ADC: m∠DAC=33°, m∠ADC=?, m∠ACD= 180 - (33 + m∠ADC). But also, incenter angle formula: m∠ACD = 90° + ½m∠AMD = 90 + 39=129. So 33 + m∠ADC +129=180 → m∠ADC=180-162=18? No, mistake earlier: m∠MAD is 66°, so m∠AMD=78°, so m∠ADM=180-66-78=36°. Then m∠ADC=½×36=18? No, wait the given 33° is m∠PAC, which is half of m∠MAD, so m∠MAD=66°, correct. Then m∠AMD= 2×m∠AMC=2×(3×11+6)=2×39=78°, correct. Then m∠ADM=180-66-78=36°, so DC bisects it, so m∠ADC=18°. Wait, earlier calculation was wrong. Let's redo step 5: 66+78=144, 180-144=36, yes, m∠ADM=36°, so m∠ADC=18°. But wait, let's use the incenter angle for ∠ADC: ∠ADC=90° - ½m∠AMD=90-39=51? No, no, the formula is that the angle between two angle bisectors is 90° + ½ the opposite angle. So ∠AMC=90+½∠ADM, so 39=90+½∠ADM → ½∠ADM= -51, which is wrong. Oh right! The correct formula is: For incenter C, ∠AMC=90° + ½∠ADM. So 3x+6=90 + ½m∠ADM. But we know ∠AMC=∠DMC, so 3x+6=8x-49, x=11, ∠AMC=39. So 39=90 + ½m∠ADM → that can't be. I mixed up the formula: The correct formula is that the angle at the incenter opposite side AM is ∠DCM=90+½∠DAM. Oh right! I had the formula backwards. Let's start over with correct logic: Since C is incenter, MC bisects ∠AMD, so ∠AMC=∠DMC, so 3x+6=8x-49 → 5x=55 →x=11, correct. ∠AMC=3(11)+6=39, so ∠AMD=78°, correct. ∠MAD is given as 2×33=66°, correct. Then ∠ADM=180-66-78=36°, correct. DC bisects ∠ADM, so ∠ADC=18°, correct. Then for ∠ADC, we can also calculate using triangle angle sum in △ADC: ∠DAC=33°, ∠ACD=180-33-18=129°, which also fits the formula ∠ACD=90+½∠AMD=90+39=129°, correct. So final answers: a)11, b)39°, c)66°, d)36°, e)18° Step 7: Verify all values with triangle angle sum and incenter properties to confirm accuracy.