Question

question 4 of 10
which descriptions from the list below accurately describe
the relationship between △ abc and △ def? check all that
apply.

a. same size
b. same shape
c. similar
d. congruent

Explanation:

To determine the relationship between \(\triangle ABC\) and \(\triangle DEF\), we analyze their angles and sides:

Step 1: Check Angles

• In \(\triangle ABC\): \(\angle A = 22^\circ\), \(\angle C = 35^\circ\), so \(\angle B = 180^\circ - 22^\circ - 35^\circ = 123^\circ\)? Wait, no, wait the diagram: Wait, \(\triangle ABC\) has angles: \(\angle A = 22^\circ\), \(\angle B = 120^\circ\)? Wait, no, the first triangle: \(A\) has \(22^\circ\), \(B\) has \(120^\circ\)? Wait, no, let's recalculate. Wait, sum of angles in a triangle is \(180^\circ\). For \(\triangle ABC\): if \(\angle A = 22^\circ\), \(\angle C = 35^\circ\), then \(\angle B = 180 - 22 - 35 = 123^\circ\)? But the second triangle \(\triangle DEF\): \(\angle E = 120^\circ\), \(\angle F = 38^\circ\), so \(\angle D = 180 - 120 - 38 = 22^\circ\). Wait, maybe I misread the angles. Wait the first triangle: \(A\) is \(22^\circ\), \(B\) is \(120^\circ\)? Wait the diagram: \(\triangle ABC\) has angle at \(A\): \(22^\circ\), angle at \(B\): \(120^\circ\)? Wait no, the first triangle: sides \(AB = 5\), \(BC = 3\), \(AC = 7\). Angles: \(\angle A = 22^\circ\), \(\angle C = 35^\circ\), so \(\angle B = 180 - 22 - 35 = 123^\circ\)? Wait the second triangle: \(DE = 10\), \(EF = 6\), \(DF = 14\). Angles: \(\angle E = 120^\circ\), \(\angle F = 38^\circ\), so \(\angle D = 22^\circ\). Wait, maybe the angles are: \(\triangle ABC\): \(\angle A = 22^\circ\), \(\angle B = 120^\circ\)? No, 22 + 120 + 35 = 177, no. Wait maybe the first triangle: \(\angle A = 22^\circ\), \(\angle B = 123^\circ\), \(\angle C = 35^\circ\). Second triangle: \(\angle D = 22^\circ\), \(\angle E = 120^\circ\)? No, 22 + 120 + 38 = 180. Wait, maybe the first triangle's angle at \(B\) is \(120^\circ\)? Wait the diagram: \(\triangle ABC\) has angle at \(B\) labeled \(120^\circ\)? Wait the user's diagram: first triangle: \(A\) with \(22^\circ\), \(B\) with \(120^\circ\)? Wait no, let's check the sides. The sides of \(\triangle ABC\) are \(AB = 5\), \(BC = 3\), \(AC = 7\). The sides of \(\triangle DEF\) are \(DE = 10\), \(EF = 6\), \(DF = 14\). Let's check the ratios: \(DE/AB = 10/5 = 2\), \(EF/BC = 6/3 = 2\), \(DF/AC = 14/7 = 2\). So the sides are in proportion (scale factor 2). Now check angles: in \(\triangle ABC\), angles: let's calculate \(\angle A\): using Law of Sines: \(\frac{BC}{\sin A} = \frac{AC}{\sin B} = \frac{AB}{\sin C}\). \(BC = 3\), \(AC = 7\), \(AB = 5\). \(\sin A / 3 = \sin B / 7 = \sin C / 5\). Sum of angles: \(A + B + C = 180^\circ\). In \(\triangle DEF\), angles: \(\angle D = 22^\circ\) (same as \(\angle A\)), \(\angle F = 38^\circ\), \(\angle E = 120^\circ\). Wait, if the sides are in proportion (scale factor 2), then the triangles are similar (by SSS similarity, since all sides are proportional). Similar triangles have the same shape (same angles) but not necessarily same size (since scale factor is 2, so \(\triangle DEF\) is larger). Congruent triangles have same size and shape (scale factor 1), but here scale factor is 2, so not congruent.

Step 2: Analyze Each Option

• Option A: Same size → No, because sides are scaled by 2, so different sizes.
• Option B: Same shape → Yes, similar triangles have the same shape (corresponding angles equal).
• Option C: Similar → Yes, because sides are in proportion (SSS similarity), so they are similar.
• Option D: Congruent → No, congruent requires same size (scale factor 1), but here scale factor is 2.

Answer:

B. Same shape, C. Similar