in triangle rst, ( mangle r>mangle s + mangle t ). which must be true of triangle rst? check all that…

in triangle rst, ( mangle r>mangle s + mangle t ). which must be true of triangle rst? check all that apply.\n( square mangle r>90^{circ} )\n( square mangle s + mangle t<90^{circ} )\n( square mangle s = mangle t )\n( square mangle r>mangle t )\n( square mangle r>mangle s )\n( square mangle s>mangle t )

in triangle rst, ( mangle r>mangle s + mangle t ). which must be true of triangle rst? check all that apply.\n( square mangle r>90^{circ} )\n( square mangle s + mangle t<90^{circ} )\n( square mangle s = mangle t )\n( square mangle r>mangle t )\n( square mangle r>mangle s )\n( square mangle s>mangle t )

Answer

Explanation:

Step1: Use the triangle angle - sum property

In any triangle ( \triangle RST), (m\angle R+m\angle S + m\angle T=180^{\circ}), so (m\angle S + m\angle T=180^{\circ}-m\angle R). Given (m\angle R>m\angle S + m\angle T). Substitute (m\angle S + m\angle T = 180^{\circ}-m\angle R) into the inequality: (m\angle R>180^{\circ}-m\angle R).

Step2: Solve the inequality for (m\angle R)

Add (m\angle R) to both sides of the inequality (m\angle R>180^{\circ}-m\angle R): (m\angle R+m\angle R>180^{\circ}), (2m\angle R>180^{\circ}). Divide both sides by 2: (m\angle R > 90^{\circ}). Since (m\angle R>m\angle S + m\angle T), and (m\angle S\geq0), (m\angle T\geq0), then (m\angle R>m\angle T) and (m\angle R>m\angle S) (because if (a>b + c) with (b\geq0) and (c\geq0), then (a>b) and (a > c)). Also, since (m\angle S + m\angle T=180^{\circ}-m\angle R) and (m\angle R>90^{\circ}), then (m\angle S + m\angle T<90^{\circ}).

Answer:

(m\angle R>90^{\circ}), (m\angle S + m\angle T<90^{\circ}), (m\angle R>m\angle T), (m\angle R>m\angle S)