Property of 30°- 60°- 90° Triangle Theorem

In the following figure, in ΔABC, ∠B = 90°, ∠C = 60°, ∠A = 30°, AC = 18 cm. Find BC.

In ΔDEF, if ∠E = 90°, then find the value of ∠D + ∠F. 

In right-angled triangle PQR, if ∠P = 60°, ∠R = 30° and PR = 12, then find the values of PQ and QR. 

Find the height of an equilateral triangle whose side is 6 units.

In the given figure, ∆PQR is an equilateral triangle. Point S is on seg QR such that QS = n\[\frac{1}{3}\] QR.

Prove that: 9 PS² = 7 PQ²

As shown in figure, LK = `6sqrt(2)` then (1) MK = ? (2) ML = ? (3) MN = ?

From the given figure, In ∆ABC, If AD ⊥ BC, ∠C = 45°, AC = `8sqrt(2)`, BD = 5, then for finding value of AD and BC, complete the following activity.

Activity: In ∆ADC, If ∠ADC = 90°, ∠C = 45°   ...(Given)

∴ ∠DAC = `square`   ...(Remaining angle of ∆ADC)

By theorem of 45° – 45° – 90° triangle,

∴ `square = 1/sqrt(2)` AC and `square = 1/sqrt(2)` AC

∴ AD =`1/sqrt(2) xx square` and DC = `1/sqrt(2) xx 8sqrt(2)`

∴ AD = 8 and DC = 8

∴ BC = BD + DC

= 5 + 8

= 13

From given figure, In ∆MNK, If ∠MNK = 90°, ∠M = 45°, MK = 6, then for finding value of MK and KN, complete the following activity.

Activity: In ∆MNK, If ∠MNK = 90°, ∠M = 45°   ...(Given)

∴ ∠K = `square`   ...(Remaining angle of ∆MNK)

By theorem of 45° – 45° – 90° triangle,

∴ `square = 1/sqrt(2)` MK and `square = 1/sqrt(2)` MK

∴ MN = `1/sqrt(2) xx square` and KN = `1/sqrt(2) xx 6`

∴ MN = `3sqrt(2)` and KN = `3sqrt(2)`