Question

1. sat/act look at △rst.

which statement is false?
ⓐ ts = tr
ⓑ m∠str < m∠trs
ⓒ tr > sr
ⓓ ts < sr
ⓔ ts + tr > sr

Explanation:

Step1: Find the measure of \(\angle R\)

Use the triangle - angle sum theorem (\(\angle T+\angle S+\angle R = 180^{\circ}\)). Given \(\angle T = 52^{\circ}\) and \(\angle S=64^{\circ}\), then \(\angle R=180-(52 + 64)=64^{\circ}\).

Step2: Analyze side - angle relationships

• Option A:

Since \(\angle S=\angle R = 64^{\circ}\), by the isosceles triangle theorem (if two angles of a triangle are equal, then the sides opposite those angles are equal), \(TS = TR\).

• Option B:

\(m\angle STR = 52^{\circ}\), \(m\angle TRS=64^{\circ}\), so \(m\angle STR

• Option C:

In \(\triangle RST\), \(\angle S = 64^{\circ}\), \(\angle T=52^{\circ}\). By the side - angle relationship (the larger angle has the longer side opposite it), since \(\angle S>\angle T\), \(TR>SR\) (because \(TR\) is opposite \(\angle S\) and \(SR\) is opposite \(\angle T\)).

• Option D:

Since \(\angle R=\angle S = 64^{\circ}\), \(TS = TR\). And from Option C, \(TR>SR\), so \(TS>SR\). This statement is false.

• Option E:

By the triangle inequality theorem, for any triangle with sides \(a,b,c\), \(a + b>c\). In \(\triangle RST\), \(TS+TR>SR\).

Answer:

D. \(TS < SR\)